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**Physics Subject Area Test**

MECHANICS: KINEMATICS

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**Motion in One DImension**

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**One Dimensional Motion**

To simplify the concept of motion, we will first consider motion that takes place in one direction. One example is the motion of a commuter train on a straight track. To measure motion, you must first choose a frame of reference. A frame of reference is a system for specifying the precise location of objects in space and time. In the train example, any station along the route.

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**Displacement ∆ x = xf -xi Displacement is a change in position.**

Displacement is not always equal to the distance traveled. The SI unit of displacement is the meter, m. ∆ x = xf -xi Displacement – final position – initial position

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**Displacement is not always equal to the distance traveled.**

Example: If a gecko starts at an initial position of 20 cm and moves to the 80 cm mark, then retreats back to the 50 cm mark as its final position, How far has the gecko traveled? What is its displacement? The gecko traveled 90 cm, but its displacement is 30 cm.

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**Postive & Negative displacement**

In general, right (east) is positive as well as upward (north) and left (west) is negative as well as downward (south).

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Average velocity Average velocity is the total displacement divided by the time interval during which the displacement occurred. In SI, the unit of velocity is meters per second abbreviated as m/s.

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Example: Consider a trip to a friend’s house 370 km to the west (negative direction) along a straight highway. If you left at 10 AM and arrived at 3 PM, what is your average velocity? This is your average. You did not travel at 74 km/h at every moment.

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**Velocity & speed Velocity is not the same as speed.**

Velocity describes motion with both direction and a numerical value (magnitude). Speed has no direction, only magnitude. Average speed is equal to the total distance traveled divided by the time interval.

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**Interpreting velocity graphically**

Consider an object whose position-time graph is not a straight line, but a curve. We obtain different average velocities depending on the time interval. The instantaneous velocity is the velocity of the object at a specific point in the object’s path The instaneous velocity can be determined by measuring the slope of the line that is tangent to that point on the diatance-vs-time graph.

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acceleration

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Changes in velocity Acceleration – Rate at which velocity changes over time An object accelerates if its speed, direction or both change. Acceleration has direction and magnitude. Acceleration is a vector quantity.

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**Acceleration has the dimensions of length divided by time squared. **

SI units are m/s2 Remember we have (m/s)/s = m/s2 Acceleration

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Example: A bus slows down with an average acceleration of -1.8 m/s2. How long does it take the bus to slow down from 9.0 m/s to a complete stop?

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Changes in Velocity Consider a train moving to the right, so that the displacement and velocity are positive. The slope of the velocity-time graph is the average acceleration. When the velocity in the positive direction is increasing, the acceleration is positive, as at A. When the velocity is constant, there is no acceleration, as at B. When the velocity in the positive direction is decreasing, the acceleration is negative, as at C.

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**Velocity & acceleration**

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**Motion with constant acceleration**

When velocity changes by the same amount during each time interval, acceleration is constant. The relationships between displacement, time, velocity, and constant acceleration are expressed by the equations shown on the next slide.

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**These equations apply to any object moving with constant acceleration**

These equations apply to any object moving with constant acceleration. These equations use the following symbols:

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**Equations for constantly accelerated straight line motion**

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Example: A racing car reaches a speed of 42 m/s. It begins a uniform negative acceleration , using its parachute and braking system, and comes to a complete rest 5.5 s later. Find the distance that the car travels during braking.

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Vectors

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**Physical Quantities are of 2 types…**

A scalar is only a magnitude (length) (Example: Temperature, time, mass) A vector has magnitude and direction (Example: displacement = 10 m East, Velocity= 50 mph west)

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**A vector will be symbolized by the “letter” with an arrow over it**

A vector will be symbolized by the “letter” with an arrow over it. The arrow indicates direction. Vectors are equal if they have the same units, magnitude, and direction. A vector can be moved anywhere parallel to itself.

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**Adding Vectors (attach)**

To add vectors they must have the same units. Tip-to -tail method put them head to tail and connect them so you end up with a triangle. Parallelogram Method- (put them tail to tail) make vectors parallel and draw a line making 2 triangles Adding Vectors (attach)

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**More Properties of Vectors**

Resultant Vector The resultant vector is the sum of a given set of vectors More Properties of Vectors

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Subtracting Vectors Tip to tail- subtract by putting vector in the opposite direction If you change the sign of a vector it is not the same vector. It is a new vector. A – B does not equal B - A

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**Components of a Vector A component is a part**

It is useful to use rectangular components These are the projections of the vector along the x- and y-axes

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**Components of a Vector, cont.**

The x-component of a vector is the projection along the x-axis The y-component of a vector is the projection along the y-axis Then,

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Useful Formulas.. The Pythagorean Theorem can only be used with right triangles!

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When its not 900 R2= A2 + B2 – 2AB(COSӨ)

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Example 1 Find the magnitude of the sum of a 15 km displacement and a 25 km displacement when the angle between them is 900 and when the angle between them is 1350.

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Example Problem Find the horizontal and vertical components of the 100m displacement of a superhero who flies from the top of a tall building at an angle of 30.00 (b) Suppose instead the superhero leaps in the other direction along a displacement vector B to the top of a flagpole where the displacement components are given Bx = -25m and BY=10.0m. Find the magnitude and direction of the displacement vector. Do this one in class… The worksheet # are different.

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Example 2 A GPS receiver indicates that your home is km and 400 north of west, but the only path through the woods leads directly north. If you follow the path 5.0 km before it opens into a field, how far, and in what direction, would you have to walk to reach your home? R= 12.39 Ө= 158’

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**Resolving a Vector Into Components**

+y The horizontal, or x-component, of A is found by Ax = A cos q. A Ay q Ax The vertical, or y-component, of A is found by Ay = A sin q. +x By the Pythagorean Theorem, Ax2 + Ay2 = A2 Every vector can be resolved using these formulas, such that A is the magnitude of A, and q is the angle the vector makes with the x-axis. Each component must have the proper “sign” according to the quadrant the vector terminates in.

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**Rx2 + Ry2 = R2 Analytical Method of Vector Addition**

1. Find the x- and y-components of each vector. Ax = A cos q = Ay = A sin q = Bx = B cos q = By = B sin q = Cx = C cos q = Cy = C sin q = Rx = Ry = 2. Sum the x-components. This is the x-component of the resultant. 3. Sum the y-components. This is the y-component of the resultant. 4. Use the Pythagorean Theorem to find the magnitude of the resultant vector. Rx2 + Ry2 = R2

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A roller coaster moves 215 ft horizontally and then rises 130 ft at an angle of above the horizontally. Next, it travels 125 ft at an angle of below the horizontal. Find the roller coaster’s displacement from its starting point to the end of this movement.

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A quarter back takes the ball from the line of scrimmage, runs backwards for 15.0 yards, then runs sideways parallel to the line of scrimmage for 15.0 yards. At this point, he throws a 60.0 yard forward pass straight downfield, perpendicular to the line of scrimmage. What is the magnitude of the football’s resultant displacement?

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**Vector Multiplication**

DOT PRODUCT scalar product A ∙ B A ∙ B = AB cosφ The product of the 2 vectors and the cosine of the angle between them

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**A ∙ B = (Axi + Ayj) (Bxi + Byj) **

= AxBx i ∙ i + AxBy i ∙ j + AyBx j ∙ i + AyBy j ∙ j i.i = j.j = k.k = 1 and i.j = j.i = i.k = k.i = j.k = k.j = 0 A ∙ B = AxBx i ∙ i + AyBy j ∙ j With 3 dimension: A ∙ B = AxBx i ∙ i + AyBy j ∙ j + AzBz k ∙ k

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**CROSS PRODUCT vector product A x B**

The product of the 2 vectors and the sine of the angle between them A x B is not the same as B x A … the direction is opposite

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**a x b = (a2b3 – a3b2 ) i + (a3b1 – a1b3 ) j + (a1b2- a2b1 ) k**

i x i = j x j = k x k = 0 i x j = k j x k = I k x i = j a x b = (a2b3 – a3b2 ) i + (a3b1 – a1b3 ) j + (a1b2- a2b1 ) k k i j

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**Two vectors in component forms are written as :**

In evaluating the product, we make use of the fact that multiplication of the same unit vectors gives the value of 0, while multiplication of two different unit vectors result in remaining vector with appropriate sign. Finally, the vector product evaluates to vector terms :

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Projectile Motion

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**A projectile is a motion in two dimensions**

Moving in the x and y direction A projectile is an object shot through the air. This occurs in a parabola curve. Toss a ball back and forth. Talk about the x and y directions of the ball.

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**(and air resistance, if any)**

projectile- any object that moves through the air or through space, acted on only by gravity (and air resistance, if any) Object dropped Object thrown at an angle Object thrown up

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**The vertical acceleration of a projectile is caused by gravity, so ay = -9.8 m/s2**

Parabolic Trajectory

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**We assume that g remains constant (g= -9.8m/s2) Neglect air resistance**

a in the x direction is 0 because gravity is not acting on it. Neglect air resistance Neglect the effects of the earths rotation

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**Projectiles launched horizontally**

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**To find how far the ball falls, you use the formula. y =viyt + 1/2gt2**

1st second- 5m After 2 seconds- 20m After 3 seconds- 45m The curved path of a projectile produced is a parabola (caused by both horizontal motion and vertical motion. It must accelerate only in the vertical direction)

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**A projectile Motion is 2 dimensions**

The projectile will experience two: Accelerations (ax= o and aY= -9.8m/s2) Velocities Displacements A projectile Motion is 2 dimensions

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**Upwardly Launched Projectiles**

When a projectile is launched at an upward angle, it follows a curved path and finally hits the ground because of gravity. The Vertical distance a cannonball falls below “imaginary path if no gravity” is the same vertical distance it would fall if it were dropped from rest & had been falling for the same amount of time.

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**How to solve these problems**

Draw a free body diagram with a coordinate system. Divide the information into x and y components Look at your formulas and decided which one(s) to use.

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**Formulas for horizontal and vertical motion of a projectile**

(Y) Vertical (X) Horizontal xf-xi = vixt + ½ axt 2 yf-yi = viyt + ½ ayt 2 vfx = vix + axt vfy = viy + ayt vfx2 - vix2 = 2ax(xf-xi) vfy2 = viy2 + 2ay (yf-yi) Objects that have been thrown will have a horizontal velocity that stays the same (no horizontal acceleration ax = 0m/s2) So vfx =vix in the second formula and third formulas under horizontal motion.

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One more formula…. This equation only works when yf and yi are both the same magnitude a = 2viy t

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**If a ball is thrown up in the air from a moving truck, where will it land? (Ignore air resistance)**

In front of the truck, behind the truck, or back in the truck

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**Where will a package land if it is released from a plane?**

Behind the plane, in front of the plane below the plane

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What is the horizontal distance covered by an arrow that was shot through the air at a 600 angle with a velocity of 55 m/s? Given 55m/s dy Solve 600 v = 55m/s vix=27.5 m/s viy=47.6m/s ax=0 ay=-9.8m/s2 t=? dx=? Vxi = cos 60(55m/s)=27.5 m/s Vix dx Viy = sin60(55m/s)=47.6m/s dx = Vix t (we need time) To find x dist: x = v0x t Need to find time first! dx = 27.5 m/s(9.7s) vfy=viy +ayt dx = m 0 = 47.6m/s m/s2 t -47.6m/s = -9.8m/st Total time in the air 4.86s x 2 = 9.7s 4.86 s =t

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**Relative Motion Frames of Reference**

Observers using different frames of reference may measure different displacements or velocities for an object in motion Relative Velocities the difference between the velocities relative to some common point

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Relative Motion: Suppose you are on a train platform as the train rushes through the station without stopping. Someone on board the train is pitching a ball, throwing it has hard as they can towards the back of the train. If the train’s speed is 60 mph and the pitcher is capable of throwing at 60 mph, what is the speed of the ball as you see it from the platform?

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Relative Motion Moving frame of reference A boat heading due north crosses a river with a speed of 10.0 km/h. The water in the river has a speed of 5.0 km/h due east. In general we have Determine the velocity of the boat. If the river is 3.0 km wide how long does it take to cross it?

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**Inelastic Collisions in 1D**

Conservation of Linear Momentum 𝑝 1𝑖 + 𝑝 2𝑖 = 𝑝 1𝑓 + 𝑝 2𝑓 Completely Inelastic Collision 𝑚 1 𝑣 𝑖 = 𝑚 1 + 𝑚 2 𝑣 𝑣= 𝑚 1 𝑚 1 + 𝑚 2 𝑣 𝑖 Velocity of Center of Mass 𝑝=𝑀 𝑣 𝑐𝑜𝑚 = 𝑚 1 + 𝑚 2 𝑣 𝑐𝑜𝑚 𝑣 𝑐𝑜𝑚 = 𝑝 𝑚 1 + 𝑚 2 = 𝑝 1𝑖 + 𝑝 2𝑖 𝑚 1 + 𝑚 2

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