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Do Now Starting from rest, a car undergoes a constant acceleration of 10. m/s/s. How far will the car travel in 3.0 s? How fast will the car move in 3.0 s?

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Do Now Starting from rest, a car undergoes a constant acceleration of 10 m/s/s. How far will the car travel in 3 s? How fast will the car move in 3 s? Given: Solution: Find:

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Do Now A basketball is dropped from rest from height of 1 m toward motion detector located on the floor. Draw x vs. t, v vs. t, and a vs. t graphs of the motion of the ball.

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**Kinematics in Two Dimensions**

Unit 4: Kinematics in Two Dimensions

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**Unit Plan Free Fall Projectile Motion**

Solving Problems Involving Projectile Motion Projectile Motion Is Parabolic

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**Aristotle (382BC-322BC)- Greek natural philosopher.**

A student of Plato and teacher of Alexander the Great Believed that more massive objects fall faster. Did not account air resistance. A detail of The School of Athens, a fresco by Raphael.

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**Galileo Galilei (1564-1642) – Italian physicist.**

Reexamined motion of falling objects Has been called the “Father of Modern Physics”(used models and experimentation) Postulated that all objects would fall with the same constant acceleration in the absence if air resistance.

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**Free Fall Freely falling objects are affected only by gravity.**

At a given location on the Earth and in absence of air resistance, all objects fall with the same constant acceleration. Acceleration due to gravity, or acceleration of free fall

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Air Resistance A feather an a coin accelerate equally when there is no air around them (in a vacuum). For compact objects the effect of air resistance is small enough to be neglected.

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**Accelerated Motion Due to Gravity**

We can choose y to be positive in the upward direction or in the downward direction. Consider motion up to be positive. acceleration due to gravity For problem solving, we will approximate On Earth, acceleration due to gravity always has downward direction(towards center of Earth).

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**Vertical Motion with Gravity**

Start with the key equations for 1-dimensional motion. Assume that the motion is only up and down. Since motion is vertical > , add the subscript y to the velocity, and substitute –g for a.

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**Object Thrown Up A rock is thrown upward with initial velocity 30 m/s.**

Time, t Velocity, Position After 0 seconds After 1 second After 2 seconds After 3 seconds After 4 seconds After 5 seconds After 6 seconds

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**Object Thrown Up A rock is thrown upward with initial velocity 30 m/s.**

Time, t Velocity, Displacement After 0 seconds 30 m/s After 1 second 20 m/s After 2 seconds 10 m/s After 3 seconds 0 m/s Reached top After 4 seconds -10 m/s After 5 seconds -20 m/s After 6 seconds -30 m/s

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**Object Thrown Up A rock is thrown upward with initial velocity 30 m/s.**

Time, t Velocity, Displacement After 0 seconds 30 m/s 0 m After 1 second 20 m/s 25 m After 2 seconds 10 m/s 40 m After 3 seconds 0 m/s Reached top 45 m After 4 seconds -10 m/s After 5 seconds -20 m/s After 6 seconds -30 m/s

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Velocity vs. Time Graph

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**Object Thrown Up. Graphs**

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Position vs. Time Graph

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**Object Thrown Up What is the instantaneous speed at the highest point?**

How does velocity change during the upward part of its motion? Decreasing from to 0. How much does its speed decrease each second? The speed decreases 10 m/s each second.

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Object Thrown Up What is the instantaneous speed of the object at points of equal elevation? The same. Are velocities same or different at points of equal elevation? Same magnitude, opposite directions. Is acceleration different when the object moving upward or downward? The same 10 m/s/s downwards.

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Dropped Object A rock is dropped from the top of the cliff. How far did it travel in 1s, 2s, and 3s?

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Dropped Object A rock is dropped from the top of the cliff. How far did it travel in 1s, 2s, and 3s? Drop Time 1 second 2 seconds 3 seconds

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**Dropping with . Find time if you know Δy.**

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Time Up = Time Down Since for the object thrown upward the motion up and down is symmetrical, you can use the same formula to find the time to go up a certain distance. If you throw a ball upwards with just enough velocity to go up a distance of 35 m, how long will it take to reach the top?

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Exercise 1 A ball is thrown upward with an initial velocity of 20 m/s. How long will it take for the ball to reach its maximum height? Given: Solution: Find:

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**Dropping With Initial Velocity Exercise 2**

A ball is thrown downward from the top of a roof with a speed of 25 m/s. Find the instantaneous velocity of the ball in 2 seconds.

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**ConcepTest 2.8b Acceleration II**

When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path? 1) both v = 0 and a = 0 2) v ¹ 0, but a = 0 3) v = 0, but a ¹ 0 4) both v ¹ 0 and a ¹ 0 5) not really sure

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**ConcepTest 2.8b Acceleration II**

When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path? 1) both v = 0 and a = 0 2) v ¹ 0, but a = 0 3) v = 0, but a ¹ 0 4) both v ¹ 0 and a ¹ 0 5) not really sure y At the top, clearly v = 0 because the ball has momentarily stopped. But the velocity of the ball is changing, so its acceleration is definitely not zero! Otherwise it would remain at rest!! Follow-up: …and the value of a is…?

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**ConcepTest 2.9a Free Fall I**

You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration? 1) its acceleration is constant everywhere 2) at the top of its trajectory 3) halfway to the top of its trajectory 4) just after it leaves your hand 5) just before it returns to your hand on the way down

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**ConcepTest 2.9a Free Fall I**

You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration? 1) its acceleration is constant everywhere 2) at the top of its trajectory 3) halfway to the top of its trajectory 4) just after it leaves your hand 5) just before it returns to your hand on the way down The ball is in free fall once it is released. Therefore, it is entirely under the influence of gravity, and the only acceleration it experiences is g, which is constant at all points.

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**ConcepTest 2.9b Free Fall II**

Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release? 1) Alice’s ball 2) it depends on how hard the ball was thrown 3) neither -- they both have the same acceleration 4) Bill’s ball v0 Bill Alice vA vB

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**ConcepTest 2.9b Free Fall II**

Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release? 1) Alice’s ball 2) it depends on how hard the ball was thrown 3) neither -- they both have the same acceleration 4) Bill’s ball Both balls are in free fall once they are released, therefore they both feel the acceleration due to gravity (g). This acceleration is independent of the initial velocity of the ball. v0 Bill Alice vA vB Follow-up: Which one has the greater velocity when they hit the ground?

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**ConcepTest 2.10a Up in the Air I**

You throw a ball upward with an initial speed of 10 m/s. Assuming that there is no air resistance, what is its speed when it returns to you? 1) more than 10 m/s 2) 10 m/s 3) less than 10 m/s 4) zero 5) need more information

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**ConcepTest 2.10a Up in the Air I**

You throw a ball upward with an initial speed of 10 m/s. Assuming that there is no air resistance, what is its speed when it returns to you? 1) more than 10 m/s 2) 10 m/s 3) less than 10 m/s 4) zero 5) need more information The ball is slowing down on the way up due to gravity. Eventually it stops. Then it accelerates downward due to gravity (again). Since a = g on the way up and on the way down, the ball reaches the same speed when it gets back to you as it had when it left.

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**ConcepTest 2.10b Up in the Air II**

Alice and Bill are at the top of a cliff of height H. Both throw a ball with initial speed v0, Alice straight down and Bill straight up. The speeds of the balls when they hit the ground are vA and vB. If there is no air resistance, which is true? 1) vA < vB 2) vA = vB 3) vA > vB 4) impossible to tell v0 Bill Alice H vA vB v0 Bill Alice H vA vB [CORRECT 5 ANSWER]

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**ConcepTest 2.10b Up in the Air II**

Alice and Bill are at the top of a cliff of height H. Both throw a ball with initial speed v0, Alice straight down and Bill straight up. The speeds of the balls when they hit the ground are vA and vB. If there is no air resistance, which is true? 1) vA < vB 2) vA = vB 3) vA > vB 4) impossible to tell v0 Bill Alice H vA vB Bill’s ball goes up and comes back down to Bill’s level. At that point, it is moving downward with v0, the same as Alice’s ball. Thus, it will hit the ground with the same speed as Alice’s ball. Follow-up: What happens if there is air resistance?

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Projectile Motion 2-D Kinematics

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A projectile is any object that is given an initial velocity or dropped and then follows a path determined entirely by the effects of gravity. Projectiles - batted baseball, a thrown football, a package dropped from an airplane, a bullet shot from a rifle. The path followed by a projectile is called its trajectory. The trajectory of a projectile is a parabola. Projectile Motion

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**Horizontally Launched Projectile**

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**Horizontal and Vertical Motion**

We can analyze projectile motion as a combination of horizontal motion with constant velocity and vertical motion with constant acceleration.

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3-5 Projectile Motion It can be understood by analyzing the horizontal and vertical motions separately.

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**Independence of Horizontal and Vertical Components**

The vertical force acts perpendicular to the horizontal motion and will not affect it since perpendicular components of motion are independent of each other. Thus, the projectile travels with a constant horizontal velocity and a downward vertical acceleration.

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**Independence of Horizontal and Vertical Motion Demo**

Two balls released simultaneously. One ball dropped freely, another projected horizontally Both balls fall the same vertical distance in equal times.

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3-5 Projectile Motion The speed in the x-direction is constant; in the y-direction the object moves with constant acceleration g. This photograph shows two balls that start to fall at the same time. The one on the right has an initial speed in the x-direction. It can be seen that vertical positions of the two balls are identical at identical times, while the horizontal position of the yellow ball increases linearly.

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Projectile Motion Vertical motion: Vertical downward acceleration: Vertical velocity is constantly changing Horizontal motion: Horizontal velocity is never changing

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**Practice Problem Vertical: Horizontal:**

The boy on a tower (h = 5m) throws a ball a distance of 20m. At what speed is the ball thrown? Given: Solution: Vertical: Horizontal: Find: t= 1.00 s

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Do Now A stone is thrown horizontally at a speed of m/s from the top of a cliff 80.0 m high. How long does it take the stone to reach the bottom of a cliff? How far from the base of the cliff does the stone strike the ground?

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Do Now A stone is thrown horizontally at a speed of m/s from the top of a cliff 80.0 m high. How long does it take the stone to reach the bottom of a cliff? How far from the base of the cliff does the stone strike the ground?

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**Given: Solution: Vertical: Horizontal: Find: a) b) **

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Conclusion A projectile is any object upon which the only force is _______, 2. Projectiles travel with a _____________ trajectory due to the influence of gravity, 3. There are __________horizontal forces acting upon projectiles and thus __________ horizontal acceleration. 4. The horizontal velocity of a projectile is ____________ 5. There is a vertical acceleration caused by gravity; its value is _______________ 6. The vertical velocity of a projectile changes by ______ m/s each second. 7. The horizontal motion of a projectile is _________________ of its vertical motion.

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Conclusion A projectile is any object upon which the only force is gravity, Projectiles travel with a parabolic trajectory due to the influence of gravity, There are no horizontal forces acting upon projectiles and thus no horizontal acceleration, The horizontal velocity of a projectile is constant (a never changing in value), There is a vertical acceleration caused by gravity; its value is 9.8 m/s/s, down, The vertical velocity of a projectile changes by 9.8 m/s each second, The horizontal motion of a projectile is independent of its vertical motion.

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Do Now A steel ball rolls with constant velocity across a tabletop m high. It rolls off and hits the ground m horizontally from the edge of the table. How fast was the ball rolling?

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**Given: Solution: Vertical: Horizontal: Find: **

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**Horizontally Launched Projectile**

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**Non-Horizontally Launched Projectile**

A cannonball is shot at an upward angle. The cannonball falls the same amount of distance in every second as it did when it was falling down.

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3-5 Projectile Motion If an object is launched at an initial angle of θ0 with the horizontal, the analysis is similar except that the initial velocity has a vertical component.

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**Horizontal and Vertical Velocity**

The horizontal component is always the same. The vertical component changes. At the top of the parabola vertical velocity = 0.

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True or False? The velocity of a projectile at its highest point is zero. False; only vertical component is zero, not velocity itself. The velocity at the highest point is equal to its horizontal component.

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**Range and Projection Angles**

Same initial speed, neglect air resistance At different angles projectiles reach different heights and have different horizontal ranges. Angles that add up to 90 degrees have the same range. The longest range has a 45 degree angle.

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**Projectile Motion With Air Resistance**

With air resistance the range is diminished. The path is not a true parabola.

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Follow-Up Question: Describe the vertical and horizontal components of a projectile launched at an angle.

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Range Formula Derive a formula for the horizontal range R of a projectile in terms of and . Solve for t: or

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Range Formula

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**3-6 Solving Problems Involving Projectile Motion**

1. Read the problem carefully, and choose the object(s) you are going to analyze. 2. Draw a diagram. 3. Choose an origin and a coordinate system. 4. Decide on the time interval; this is the same in both directions, and includes only the time the object is moving with constant acceleration g. 5. Examine the x and y motions separately. 6. List know and unknown quantities. 7. Plan how you will proceed. Use the appropriate equations; you may have to combine some of them.

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**Projectile Launched at an Angle Problem**

You shoot a rocket at an angle of 40.0°relative to the horizontal. The rocket has an initial speed of 30.0 m/s. a)What are the horizontal and vertical components of the rockets initial velocity? b) After 1.00 second of flight, how high is the rocket? c) After 1.00 second of flight, how far horizontally has the rocket traveled? d) How long will it take the rocket to travel to its highest point?

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Given: Solution: Find:

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**b) Given: Solution: Find: Δy-? **

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**c) Given: Solution: Find: Δx-? **

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**d) Given: Solution: Find: t-? **

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**ConcepTest 3.10a Shoot the Monkey I**

You are trying to hit a friend with a water balloon. He is sitting in the window of his dorm room directly across the street. You aim straight at him and shoot. Just when you shoot, he falls out of the window! Does the water balloon hit him? 1) yes, it hits 2) maybe – it depends on the speed of the shot 3) no, it misses 4) the shot is impossible 5) not really sure [CORRECT 5 ANSWER] Assume that the shot does have enough speed to reach the dorm across the street.

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**ConcepTest 3.10a Shoot the Monkey I**

You are trying to hit a friend with a water balloon. He is sitting in the window of his dorm room directly across the street. You aim straight at him and shoot. Just when you shoot, he falls out of the window! Does the water balloon hit him? 1) yes, it hits 2) maybe – it depends on the speed of the shot 3) no, it misses 4) the shot is impossible 5) not really sure Your friend falls under the influence of gravity, just like the water balloon. Thus, they are both undergoing free fall in the y-direction. Since the slingshot was accurately aimed at the right height, the water balloon will fall exactly as your friend does, and it will hit him!! [CORRECT 5 ANSWER] Assume that the shot does have enough speed to reach the dorm across the street.

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**ConcepTest 3.10b Shoot the Monkey II**

You’re on the street, trying to hit a friend with a water balloon. He sits in his dorm room window above your position. You aim straight at him and shoot. Just when you shoot, he falls out of the window! Does the water balloon hit him?? 1) yes, it hits 2) maybe – it depends on the speed of the shot 3) the shot is impossible 4) no, it misses 5) not really sure [CORRECT 5 ANSWER] Assume that the shot does have enough speed to reach the dorm across the street.

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**ConcepTest 3.10b Shoot the Monkey II**

You’re on the street, trying to hit a friend with a water balloon. He sits in his dorm room window above your position. You aim straight at him and shoot. Just when you shoot, he falls out of the window! Does the water balloon hit him?? 1) yes, it hits 2) maybe – it depends on the speed of the shot 3) the shot is impossible 4) no, it misses 5) not really sure This is really the same situation as before!! The only change is that the initial velocity of the water balloon now has a y-component as well. But both your friend and the water balloon still fall with the same acceleration -- g !! [CORRECT 5 ANSWER] Assume that the shot does have enough speed to reach the dorm across the street.

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**ConcepTest 3.10c Shoot the Monkey III**

You’re on the street, trying to hit a friend with a water balloon. He sits in his dorm room window above your position and is aiming at you with HIS water balloon! You aim straight at him and shoot and he does the same in the same instant. Do the water balloons hit each other? 1) yes, they hit 2) maybe – it depends on the speeds of the shots 3) the shots are impossible 4) no, they miss 5) not really sure

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**ConcepTest 3.10c Shoot the Monkey III**

You’re on the street, trying to hit a friend with a water balloon. He sits in his dorm room window above your position and is aiming at you with HIS water balloon! You aim straight at him and shoot and he does the same in the same instant. Do the water balloons hit each other? 1) yes, they hit 2) maybe – it depends on the speeds of the shots 3) the shots are impossible 4) no, they miss 5) not really sure This is still the same situation!! Both water balloons are aimed straight at each other and both still fall with the same acceleration -- g !! Follow-up: When would they NOT hit each other?

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**ConcepTest 3.4b Firing Balls II**

Now the cart is being pulled along a horizontal track by an external force (a weight hanging over the table edge) and accelerating. It fires a ball straight out of the cannon as it moves. After it is fired, what happens to the ball? 1) it depends upon how much the track is tilted 2) it falls behind the cart 3) it falls in front of the cart 4) it falls right back into the cart 5) it remains at rest

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**ConcepTest 3.4b Firing Balls II**

Now the cart is being pulled along a horizontal track by an external force (a weight hanging over the table edge) and accelerating. It fires a ball straight out of the cannon as it moves. After it is fired, what happens to the ball? 1) it depends upon how much the track is tilted 2) it falls behind the cart 3) it falls in front of the cart 4) it falls right back into the cart 5) it remains at rest Now the acceleration of the cart is completely unrelated to the ball. In fact, the ball does not have any horizontal acceleration at all (just like the first question), so it will lag behind the accelerating cart once it is shot out of the cannon.

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**ConcepTest 3.4c Firing Balls III**

The same small cart is now rolling down an inclined track and accelerating. It fires a ball straight out of the cannon as it moves. After it is fired, what happens to the ball? 1) it depends upon how much the track is tilted 2) it falls behind the cart 3) it falls in front of the cart 4) it falls right back into the cart 5) it remains at rest

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**ConcepTest 3.4c Firing Balls III**

The same small cart is now rolling down an inclined track and accelerating. It fires a ball straight out of the cannon as it moves. After it is fired, what happens to the ball? 1) it depends upon how much the track is tilted 2) it falls behind the cart 3) it falls in front of the cart 4) it falls right back into the cart 5) it remains at rest Because the track is inclined, the cart accelerates. However, the ball has the same component of acceleration along the track as the cart does! This is essentially the component of g acting parallel to the inclined track. So the ball is effectively accelerating down the incline, just as the cart is, and it falls back into the cart.

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**3-6 Solving Problems Involving Projectile Motion**

Projectile motion is motion with constant acceleration in two dimensions, where the acceleration is g and is down.

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**3-6 Solving Problems Involving Projectile Motion**

Read the problem carefully, and choose the object(s) you are going to analyze. Draw a diagram. Choose an origin and a coordinate system. Decide on the time interval; this is the same in both directions, and includes only the time the object is moving with constant acceleration g. Examine the x and y motions separately. List know and unknown quantities. Plan how you will proceed. Use the appropriate equations; you may have to combine some of them.

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**3-7 Projectile Motion Is Parabolic**

In order to demonstrate that projectile motion is parabolic, we need to write y as a function of x. When we do, we find that it has the form: This is indeed the equation for a parabola.

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Do Now While skiing, Ellen encounters an icy bump, which she leaves horizontally at 12.0 m/s. How far out , horizontally from her starting point will Ellen land if she drops a distance of 7.00 m in the fall?

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**Dropping With Initial Velocity Exercise 1**

A ball is thrown downward from the top of a roof with a speed of 25 m/s. Find the instantaneous velocity of the ball in 2 seconds.

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Average Speed During the span of a second time interval a falling object begins at -10 m/s and ends at m/s . What is the average speed of the object during this 1-second interval? What is its acceleration? The acceleration is -10 m/s.

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$40 40 m/s $20+$10x(3s)=$50 20+10x3=50 m/s $50-10x(time)=0 time=5s 5s 0 m/s 10 m/s, 20 m/s

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125 m 105m a. 30 m/s b. c. 45 m 4.

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Problem 1. A long jumper leaves the ground with an initial velocity of 12 m/s at an angle of 28-degrees above the horizontal. Determine the time of flight, the horizontal distance, and the peak height of the long-jumper. Given:

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