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**Projectiles and Newton’s 2nd Law**

Chapters 5 & 6 Conceptual Physics

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**Projectile motion can be described by the horizontal and vertical components of motion.**

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**In the previous chapter we studied simple straight-line motion—linear motion.**

Now we extend these ideas to nonlinear motion—motion along a curved path. Throw a baseball and the path it follows is a combination of constant-velocity horizontal motion and accelerated vertical motion.

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**5.1 Vector and Scalar Quantities**

A vector quantity includes both magnitude and direction, but a scalar quantity includes only magnitude.

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5.2 Velocity Vectors The resultant of two perpendicular vectors is the diagonal of a rectangle constructed with the two vectors as sides.

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5.2 Velocity Vectors The airplane’s velocity relative to the ground depends on the airplane’s velocity relative to the air and on the wind’s velocity.

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**The resulting speed of the airplane is 60 km/h.**

5.2 Velocity Vectors The velocity of something is often the result of combining two or more other velocities. If a small airplane is flying north at 80 km/h relative to the surrounding air and a tailwind blows north at a velocity of 20 km/h, the plane travels 100 kilometers in one hour relative to the ground below. What if the plane flies into the wind rather than with the wind? The velocity vectors are now in opposite directions. The resulting speed of the airplane is 60 km/h.

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5.2 Velocity Vectors Now consider an 80-km/h airplane flying north caught in a strong crosswind of 60 km/h blowing from west to east. The plane’s speed relative to the ground can be found by adding the two vectors. The result of adding these two vectors, called the resultant, is the diagonal of the rectangle described by the two vectors.

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5.2 Velocity Vectors An 80-km/h airplane flying in a 60-km/h crosswind has a resultant speed of 100 km/h relative to the ground.

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5.2 Velocity Vectors The 3-unit and 4-unit vectors at right angles add to produce a resultant vector of 5 units, at 37° from the horizontal.

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5.2 Velocity Vectors The diagonal of a square is , or 1.414, times the length of one of its sides.

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5.2 Velocity Vectors think! Suppose that an airplane normally flying at 80 km/h encounters wind at a right angle to its forward motion—a crosswind. Will the airplane fly faster or slower than 80 km/h?

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5.2 Velocity Vectors think! Suppose that an airplane normally flying at 80 km/h encounters wind at a right angle to its forward motion—a crosswind. Will the airplane fly faster or slower than 80 km/h? Answer: A crosswind would increase the speed of the airplane and blow it off course by a predictable amount.

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5.3 Components of Vectors The perpendicular components of a vector are independent of each other.

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5.3 Components of Vectors Often we will need to change a single vector into an equivalent set of two component vectors at right angles to each other: Any vector can be “resolved” into two component vectors at right angles to each other. Two vectors at right angles that add up to a given vector are known as the components of the given vector. The process of determining the components of a vector is called resolution.

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5.3 Components of Vectors A ball’s velocity can be resolved into horizontal and vertical components.

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5.3 Components of Vectors Vectors X and Y are the horizontal and vertical components of a vector V.

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5.4 Projectile Motion The horizontal component of motion for a projectile is just like the horizontal motion of a ball rolling freely along a level surface without friction. The vertical component of a projectile’s velocity is like the motion for a freely falling object.

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5.4 Projectile Motion A projectile is any object that moves through the air or space, acted on only by gravity (and air resistance, if any). A cannonball shot from a cannon, a stone thrown into the air, a ball rolling off the edge of a table, a spacecraft circling Earth—all of these are examples of projectiles.

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**Projectile motion can be separated into components. **

Roll a ball along a horizontal surface, and its velocity is constant because no component of gravitational force acts horizontally. Drop it, and it accelerates downward and covers a greater vertical distance each second.

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**Each component is independent of the other. **

5.4 Projectile Motion Most important, the horizontal component of motion for a projectile is completely independent of the vertical component of motion. Each component is independent of the other. Their combined effects produce the variety of curved paths that projectiles follow.

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**5.5 Projectiles Launched Horizontally**

The downward motion of a horizontally launched projectile is the same as that of free fall.

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**5.5 Projectiles Launched Horizontally**

A strobe-light photo of two balls released simultaneously–one ball drops freely while the other one is projected horizontally. Demo!!! Ring stand. Wooden marchmallo shooter. Shoot the Monkey!

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**5.5 Projectiles Launched Horizontally**

There are two important things to notice in the photo of two balls falling simultaneously: The ball’s horizontal component of motion remains constant. Gravity acts only downward, so the only acceleration of the ball is downward. Both balls fall the same vertical distance in the same time. The vertical distance fallen has nothing to do with the horizontal component of motion.

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**5.5 Projectiles Launched Horizontally**

The ball moves the same horizontal distance in the equal time intervals because no horizontal component of force is acting on it. The path traced by a projectile accelerating in the vertical direction while moving at constant horizontal velocity is a parabola. When air resistance is small enough to neglect, the curved paths are parabolic.

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**5.5 Projectiles Launched Horizontally**

think! At the instant a horizontally pointed cannon is fired, a cannonball held at the cannon’s side is released and drops to the ground. Which cannonball strikes the ground first, the one fired from the cannon or the one dropped?

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**5.6 Projectiles Launched at an Angle**

The vertical distance a projectile falls below an imaginary straight-line path increases continually with time and is equal to 5t2 meters.

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**5.6 Projectiles Launched at an Angle**

No matter the angle at which a projectile is launched, the vertical distance of fall beneath the idealized straight-line path (dashed straight lines) is the same for equal times.

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**5.6 Projectiles Launched at an Angle**

The dashed straight lines show the ideal trajectories of the stones if there were no gravity. Notice that the vertical distance that the stone falls beneath the idealized straight-line paths is the same for equal times. This vertical distance is independent of what’s happening horizontally.

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**5.6 Projectiles Launched at an Angle**

With no gravity the projectile would follow the straight-line path (dashed line). But because of gravity it falls beneath this line the same vertical distance it would fall if it were released from rest.

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**5.6 Projectiles Launched at an Angle**

With no gravity the projectile would follow the straight-line path (dashed line). But because of gravity it falls beneath this line the same vertical distance it would fall if it were released from rest.

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**5.6 Projectiles Launched at an Angle**

With no gravity the projectile would follow the straight-line path (dashed line). But because of gravity it falls beneath this line the same vertical distance it would fall if it were released from rest.

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**5.6 Projectiles Launched at an Angle**

If there were no gravity the cannonball would follow the straight-line path shown by the dashed line. The vertical distance it falls beneath any point on the dashed line is the same vertical distance it would fall if it were dropped from rest:

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**5.6 Projectiles Launched at an Angle**

The velocity of a projectile is shown at various points along its path. Notice that the vertical component changes while the horizontal component does not. Air resistance is neglected.

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**Range 5.6 Projectiles Launched at an Angle**

The angle at which the projectile is launched affects the distance that it travels.

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**Both projectiles have the same launching speed. **

5.6 Projectiles Launched at an Angle Both projectiles have the same launching speed. The initial velocity vector has a greater vertical component than when the projection angle is less. This greater component results in a higher path. The horizontal component is less, so the range is less.

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**Horizontal Ranges 5.6 Projectiles Launched at an Angle**

Projectiles that are launched at the same speed but at different angles reach different heights (altitude) above the ground. They also travel different horizontal distances, that is, they have different horizontal ranges.

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**5.6 Projectiles Launched at an Angle**

The paths of projectiles launched at the same speed but at different angles. The paths neglect air resistance.

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**Maximum range is usually attained at an angle of 45°.**

5.6 Projectiles Launched at an Angle The same range is obtained for two different projection angles—angles that add up to 90°. An object thrown into the air at an angle of 60° will have the same range as at 30° with the same speed. Maximum range is usually attained at an angle of 45°.

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**5.6 Projectiles Launched at an Angle**

Speed Without air resistance, a projectile will reach maximum height in the same time it takes to fall from that height to the ground. The deceleration due to gravity going up is the same as the acceleration due to gravity coming down. The projectile hits the ground with the same speed it had when it was projected upward from the ground.

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**The time to go up equals the time to come down.**

5.6 Projectiles Launched at an Angle Without air resistance, the speed lost while the cannonball is going up equals the speed gained while it is coming down. The time to go up equals the time to come down.

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**5.6 Projectiles Launched at an Angle**

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**5.6 Projectiles Launched at an Angle**

think! A projectile is launched at an angle into the air. Neglecting air resistance, what is its vertical acceleration? Its horizontal acceleration?

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**5.6 Projectiles Launched at an Angle**

think! A projectile is launched at an angle into the air. Neglecting air resistance, what is its vertical acceleration? Its horizontal acceleration? Answer: Its vertical acceleration is g because the force of gravity is downward. Its horizontal acceleration is zero because no horizontal force acts on it.

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**Which of these expresses a vector quantity? 10 kg 10 kg to the north **

Assessment Questions Which of these expresses a vector quantity? 10 kg 10 kg to the north 10 m/s 10 m/s to the north

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**Which of these expresses a vector quantity? 10 kg 10 kg to the north **

Assessment Questions Which of these expresses a vector quantity? 10 kg 10 kg to the north 10 m/s 10 m/s to the north Answer: D

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Assessment Questions An ultra-light aircraft traveling north at 40 km/h in a 30-km/h crosswind (at right angles) has a groundspeed of 30 km/h. 40 km/h. 50 km/h. 60 km/h.

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Assessment Questions An ultra-light aircraft traveling north at 40 km/h in a 30-km/h crosswind (at right angles) has a groundspeed of 30 km/h. 40 km/h. 50 km/h. 60 km/h. Answer: C

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**A ball launched into the air at 45° to the horizontal initially has **

Assessment Questions A ball launched into the air at 45° to the horizontal initially has equal horizontal and vertical components. components that do not change in flight. components that affect each other throughout flight. a greater component of velocity than the vertical component.

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**A ball launched into the air at 45° to the horizontal initially has **

Assessment Questions A ball launched into the air at 45° to the horizontal initially has equal horizontal and vertical components. components that do not change in flight. components that affect each other throughout flight. a greater component of velocity than the vertical component. Answer: A

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**opposite to the force of gravity. at right angles.**

Assessment Questions When no air resistance acts on a fast-moving baseball, its acceleration is downward, g. due to a combination of constant horizontal motion and accelerated downward motion. opposite to the force of gravity. at right angles.

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**opposite to the force of gravity. at right angles. Answer: A**

Assessment Questions When no air resistance acts on a fast-moving baseball, its acceleration is downward, g. due to a combination of constant horizontal motion and accelerated downward motion. opposite to the force of gravity. at right angles. Answer: A

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Assessment Questions When no air resistance acts on a projectile, its horizontal acceleration is g. at right angles to g. upward, g. zero.

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Assessment Questions When no air resistance acts on a projectile, its horizontal acceleration is g. at right angles to g. upward, g. zero. Answer: D

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**10 m/s for every second in the air. the same as the time going upward. **

Assessment Questions Without air resistance, the time for a vertically tossed ball to return to where it was thrown is 10 m/s for every second in the air. the same as the time going upward. less than the time going upward. more than the time going upward.

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**10 m/s for every second in the air. the same as the time going upward. **

Assessment Questions Without air resistance, the time for a vertically tossed ball to return to where it was thrown is 10 m/s for every second in the air. the same as the time going upward. less than the time going upward. more than the time going upward. Answer: B

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**An object accelerates when a net force acts on it.**

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**6.1 Force Causes Acceleration**

Unbalanced forces acting on an object cause the object to accelerate.

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**acceleration ~ net force**

6.1 Force Causes Acceleration Recall from the previous chapter that the combination of forces acting on an object is the net force. Acceleration depends on the net force. To increase the acceleration of an object, you must increase the net force acting on it. An object’s acceleration is directly proportional to the net force acting on it: acceleration ~ net force (The symbol ~ stands for “is directly proportional to.”)

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**6.1 Force Causes Acceleration**

Kick a football and it neither remains at rest nor moves in a straight line.

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**6.2 Mass Resists Acceleration**

For a constant force, an increase in the mass will result in a decrease in the acceleration.

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**6.2 Mass Resists Acceleration**

Push on an empty shopping cart. Then push equally hard on a heavily loaded shopping cart. The loaded shopping cart will accelerate much less than the empty cart. Acceleration depends on the mass being pushed.

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**6.2 Mass Resists Acceleration**

The same force applied to twice as much mass results in only half the acceleration. The acceleration is inversely proportional to the mass. Inversely means that the two values change in opposite directions. As the denominator increases, the whole quantity decreases by the same factor.

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**6.2 Mass Resists Acceleration**

The acceleration produced depends on the mass that is pushed.

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6.3 Newton’s Second Law Newton’s second law states that the acceleration produced by a net force on an object is directly proportional to the magnitude of the net force, is in the same direction as the net force, and is inversely proportional to the mass of the object.

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6.3 Newton’s Second Law By using consistent units, such as newtons (N) for force, kilograms (kg) for mass, and meters per second squared (m/s2) for acceleration, we get the exact equation: If a is acceleration, F is net force, and m is mass,

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6.3 Newton’s Second Law The acceleration is equal to the net force divided by the mass. If the net force acting on an object doubles, its acceleration is doubled. If the mass is doubled, then acceleration will be halved. If both the net force and the mass are doubled, the acceleration will be unchanged.

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**6.3 Newton’s Second Law think!**

If a car can accelerate at 2 m/s2, what acceleration can it attain if it is towing another car of equal mass?

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**6.3 Newton’s Second Law think!**

If a car can accelerate at 2 m/s2, what acceleration can it attain if it is towing another car of equal mass? Answer: The same force on twice the mass produces half the acceleration, or 1 m/s2.

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**6.3 Newton’s Second Law do the math!**

A car has a mass of 1000 kg. What is the acceleration produced by a force of 2000 N?

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**6.3 Newton’s Second Law do the math!**

A car has a mass of 1000 kg. What is the acceleration produced by a force of 2000 N?

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**6.3 Newton’s Second Law do the math!**

If the force is 4000 N, what is the acceleration?

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**6.3 Newton’s Second Law do the math!**

If the force is 4000 N, what is the acceleration? Doubling the force on the same mass simply doubles the acceleration.

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**6.3 Newton’s Second Law do the math!**

How much force, or thrust, must a 30,000-kg jet plane develop to achieve an acceleration of 1.5 m/s2?

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**6.3 Newton’s Second Law do the math!**

How much force, or thrust, must a 30,000-kg jet plane develop to achieve an acceleration of 1.5 m/s2? Arrange Newton’s second law to read: force = mass × acceleration F = ma = (30,000 kg)(1.5 m/s2) = 45,000 kg•m/s2 = 45,000 N

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6.4 Friction The force of friction between the surfaces depends on the kinds of material in contact and how much the surfaces are pressed together.

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**6.4 Friction Friction is a force and affects motion:**

Friction acts on materials that are in contact with each other. It always acts in a direction to oppose relative motion. When two solid objects come into contact, the friction is mainly due to irregularities in the two surfaces.

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6.4 Friction Rubber against concrete produces more friction than steel against steel, so concrete road dividers have replaced steel rails. The friction produced by a tire rubbing against the concrete is more effective in slowing the car than the friction produced by a steel car body sliding against a steel rail.

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6.4 Friction A concrete road divider has a better design than a steel road divider for slowing an out-of-control, sideswiping car. The concrete divider is wider at the bottom to ensure that the tire will make contact with the divider before the steel car body does.

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6.4 Friction Both liquids and gases are called fluids because they flow. Fluid friction occurs as an object pushes aside the fluid it is moving through. The friction of liquids is appreciable, even at low speeds. Air resistance is the friction acting on something moving through air.

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6.4 Friction When friction is present, an object may move with a constant velocity even when an outside force is applied to it. In such a case, the friction force just balances the applied force. The net force is zero, so there is no acceleration. A diagram showing all the forces acting on an object is called a free-body diagram.

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6.4 Friction The direction of the force of friction always opposes the direction of motion. a. Push the crate to the right and friction acts toward the left.

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6.4 Friction The direction of the force of friction always opposes the direction of motion. a. Push the crate to the right and friction acts toward the left. b. The sack falls downward and air friction acts upward.

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6.4 Friction think! Two forces act on a book resting on a table: its weight and the support force from the table. Does a force of friction act as well?

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6.4 Friction think! Two forces act on a book resting on a table: its weight and the support force from the table. Does a force of friction act as well? Answer: No, not unless the book tends to slide or does slide across the table. Friction forces occur only when an object tends to slide or is sliding.

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**6.5 Applying Force-Pressure**

For a constant force, an increase in the area of contact will result in a decrease in the pressure.

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**6.5 Applying Force-Pressure**

The amount of force per unit of area is called pressure. When the force is perpendicular to the surface area, P is the pressure and A is the area over which the force acts. Pressure is measured in newtons per square meter, or pascals (Pa). One newton per square meter is equal to one pascal.

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**6.5 Applying Force-Pressure**

The force of the book on the table is the same. The upright book exerts the same force, but greater pressure, against the supporting surface.

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**6.5 Applying Force-Pressure**

You exert more pressure against the ground when you stand on one foot than when you stand on both feet due to the decreased area of contact. The smaller the area supporting a given force, the greater the pressure on that surface.

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**6.5 Applying Force-Pressure**

The driving force per nail is not enough to puncture the skin. CAUTION: Do not attempt this on your own! Seat of nails demos

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6.6 Free Fall Explained All freely falling objects fall with the same acceleration because the net force on an object is only its weight, and the ratio of weight to mass is the same for all objects.

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6.6 Free Fall Explained Galileo showed that falling objects accelerate equally, regardless of their masses. This is strictly true if air resistance is negligible, that is, if the objects are in free fall. It is approximately true when air resistance is very small compared with the mass of the falling object.

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6.6 Free Fall Explained In Galileo’s famous demonstration, a 10-kg cannonball and a 1-kg stone strike the ground at practically the same time. This experiment demolished the Aristotelian idea that an object that weighs ten times as much as another should fall ten times faster than the lighter object.

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6.6 Free Fall Explained Recall that mass (a quantity of matter) and weight (the force due to gravity) are proportional. A 10-kg cannonball experiences 10 times as much gravitational force (weight) as a 1-kg stone. Newton’s second law tells us to consider the mass as well. Ten times as much force acting on ten times as much mass produces the same acceleration.

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6.6 Free Fall Explained F stands for the force (weight) acting on the cannonball, and m stands for the correspondingly large mass of the cannonball. The small F and m stand for the weight and mass of the stone. The ratio of weight to mass is the same for these or any objects. All freely falling objects undergo the same acceleration at the same place on Earth.

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6.6 Free Fall Explained The ratio of weight (F) to mass (m) is the same for the 10-kg cannonball and the 1-kg stone.

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6.6 Free Fall Explained The weight of a 1-kg stone is 10 N at Earth’s surface. Using Newton’s second law, the acceleration of the stone is The weight of a 10-kg cannonball is 100 N at Earth’s surface and the acceleration of the cannonball is

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**6.7 Falling and Air Resistance**

The air resistance force an object experiences depends on the object’s speed and area.

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**6.7 Falling and Air Resistance**

A feather and a coin fall with equal accelerations in a vacuum, but very unequally in the presence of air. When falling in air, the coin falls quickly while the feather flutters to the ground. The force due to air resistance diminishes the net force acting on the falling objects.

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**6.7 Falling and Air Resistance**

Speed and Area You experience the force due to air resistance when you stick your hand out of the window of a moving car. If the car moves faster, the force on your hand increases. If instead of just your hand, you hold your physics book out the window with the large side facing forward, the air resistance force is much larger than on your hand at the same speed.

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**6.7 Falling and Air Resistance**

Air resistance force ~ speed × frontal area The expression shows that the air resistance force is directly proportional to the speed and frontal area of an object.

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**6.7 Falling and Air Resistance**

Terminal Speed Terminal speed is the speed at which the acceleration of a falling object is zero because friction balances the weight. Terminal velocity is terminal speed together with the direction of motion.

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**6.7 Falling and Air Resistance**

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**6.7 Falling and Air Resistance**

Sky divers reach terminal speed when air resistance equals weight.

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**6.7 Falling and Air Resistance**

A falling feather reaches its terminal speed quite quickly. Its area is large relative to its very small weight so air resistance has a large effect on the feather’s motion. A coin has a relatively small area compared to its weight, so the coin will have to fall faster to reach its terminal speed.

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**6.7 Falling and Air Resistance**

The terminal speed for a sky diver varies from about 150 to 200 km/h, depending on the weight and orientation of the body. A heavier person will attain a greater terminal speed than a lighter person. Body orientation also makes a difference. More air is encountered when the body is spread out and surface area is increased.

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**6.7 Falling and Air Resistance**

The flying squirrel increases its area by spreading out. This increases air resistance and decreases the speed of its fall.

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**6.7 Falling and Air Resistance**

Terminal speed can be controlled by variations in body orientation. A heavy sky diver and a light sky diver can remain in close proximity to each other if the heavy person spreads out like a flying squirrel while the light person falls head or feet first. A parachute greatly increases air resistance, and cuts the terminal speed down to 15 to 25 km/h, slow enough for a safe landing.

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**6.7 Falling and Air Resistance**

This stroboscopic photo shows a golf ball and a foam ball falling in air. The heavier golf ball is more effective in overcoming air resistance, so its acceleration is greater.

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**6.7 Falling and Air Resistance**

think! If a heavy person and a light person open their parachutes together at the same altitude and each wears the same size parachute, who will reach the ground first? Answer: The heavy person will reach the ground first. Like a feather, the light person reaches terminal speed sooner, while the heavy person continues to accelerate until a greater terminal speed is reached.

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**Assessment Questions An object will accelerate when SF = 0.**

it is unbalanced. it is pushed or pulled with a net force. its mass increases.

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**Assessment Questions An object will accelerate when SF = 0.**

it is unbalanced. it is pushed or pulled with a net force. its mass increases. Answer: C

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Assessment Questions When a net force acts on an object, its acceleration depends on the object’s initial speed. mass. volume. weight.

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Assessment Questions When a net force acts on an object, its acceleration depends on the object’s initial speed. mass. volume. weight. Answer: B

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Assessment Questions A cart is pushed and undergoes a certain acceleration. Consider how the acceleration would compare if it were pushed with twice the net force while its mass increased by four. Then its acceleration would be one quarter. half. twice. the same.

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Assessment Questions A cart is pushed and undergoes a certain acceleration. Consider how the acceleration would compare if it were pushed with twice the net force while its mass increased by four. Then its acceleration would be one quarter. half. twice. the same. Answer: B

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Assessment Questions Friction is a force like any other force and affects motion. Friction occurs in solids sliding over one another. fluids. air. all of these

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Assessment Questions Friction is a force like any other force and affects motion. Friction occurs in solids sliding over one another. fluids. air. all of these Answer: D

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Assessment Questions When you stand on one foot instead of two, the pressure you exert on the ground is half. the same. twice. quadruple.

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Assessment Questions When you stand on one foot instead of two, the pressure you exert on the ground is half. the same. twice. quadruple. Answer: C

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Assessment Questions The reason a 20-kg rock falls no faster than a 10-kg rock in free fall is that air resistance is negligible. the force of gravity on both is the same. their speeds are the same. the force/mass ratio is the same.

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Assessment Questions The reason a 20-kg rock falls no faster than a 10-kg rock in free fall is that air resistance is negligible. the force of gravity on both is the same. their speeds are the same. the force/mass ratio is the same. Answer: D

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Assessment Questions Kevin and Suzanne go sky diving. Kevin is heavier than Suzanne, but both use the same size parachute. Kevin has a greater terminal speed compared with Suzanne because he has to fall faster for air resistance to match his weight. gravity acts on him more. he has greater air resistance. he has weaker terminal velocity.

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Assessment Questions Kevin and Suzanne go sky diving. Kevin is heavier than Suzanne, but both use the same size parachute. Kevin has a greater terminal speed compared with Suzanne because he has to fall faster for air resistance to match his weight. gravity acts on him more. he has greater air resistance. he has weaker terminal velocity. Answer: A

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