Physics



AP Physics 2: Kinematics—Two Dimension Name __________________________

A. Vectors

1. vectors and scalars

a. d, v and a have magnitude (how much) and direction (which way) ∴ vectors

b. t, m, distance, speed only have magnitude ∴ scalars

c. arrow is used to symbolize a vector

1. arrow points in the vector’s direction

2. magnitude ∝ arrow's length

2. addition of vectors—tail to tip method

vectors are laid out tail to tip; the sum (resultant) equals the length and angle of the line that connects the tail of the first vector to the tip of the last vector

B

R = A + B

A

3. addition of vectors—component method

|an x-y coordinate system is established and θ is measured |

|counterclockwise from +x-axis (0o) |

|y = 90o |

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|B |

|θB |

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|A |

|θA |

|x = 0o |

|x-component and y-component for each vector are calculated (R = |

|magnitude; θ = direction) |

|cosθ = adjacent/hypotenuse ∴ Rx = Rcosθ |

|sinθ = opposite/hypotenuse ∴ Ry = Rsinθ |

|y Rx = Ax + Bx Bx= BcosθB |

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|B By = BsinθB |

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|Ry = Ay + By R |

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|Ay = AsinθA |

|A |

|x |

|Ax = AcosθA |

|Rx = Ax + Bx = AcosθA + BcosθB |

|Ry = Ay + By = AsinθA + BsinθB |

|R = (Rx2 + Ry2)½ |

|tanθ = Ry/Rx ∴ θ = tan-1(Ry/Rx) |

|add 180o to θ when Rx is negative |

4. relative motion

a. observed velocity of an object depends on the motion of the observer relative to the object

b. observed acceleration is the same for any non-accelerating observer

c. example: a boat is traveling vboat with respect to the water, but the water current is vwater with respect to the Earth ∴ the boat velocity with respect to the Earth is vector sum: v = vboat + vwater

river current → vboat vboat + vwater

vwater

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B. Projectile Motion

1. projectile's horizontal velocity is constant

2. projectile's vertical velocity is immediately affected by the downward acceleration of gravity

3. solving projectile motion problems

a. general solution

|determine vxo = vocosθ and vyo = vosinθ |

|complete the "y" row in the data chart for all values in the vertical |

|direction (↓ is negative) |

|complete the "x" row in the data chart for all values in the horizontal |

|direction (vxo = vx) |

|time is the same for y and x directions |

|direction |

|d |

|vo |

|vt |

|a |

|t |

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|y |

|dy |

|vyo |

|vyt |

|-10 m/s2 |

|t |

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|x |

|dx |

|vx |

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|solve for unknown in the vertical direction with |

|dy = vyot + ½gt2 |

|vyt = vyo + gt |

|vyt2 = vyo2 + 2gdy |

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|no vyt |

|no dy |

|no t |

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|solve for unknown in the x direction with dx = vxt |

b. projectile launched horizontally from height h

|data chart |

|direction |

|d |

|vo |

|vt |

|a |

|t |

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|y |

|-h |

|0 |

|vyt |

|-10 m/s2 |

|t |

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|x |

|dx |

|vx = vo |

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|solve for unknown in the vertical direction with |

|h = ½gt2 |

|vyt = gt |

|vyt2 = 2gh |

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|solve for unknown in the x direction with dx = vxt |

c. projectile launched at angle θo from height 0

|determine vxo = vocosθo and vyo = vosinθo |

|data chart—highest point |

|direction |

|d |

|vo |

|vt |

|a |

|t |

| |

|y |

|dy |

|vyo |

|0 |

|-10 m/s2 |

|t |

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|x |

|dx |

|vx |

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| |

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|solve for unknown in the vertical direction with |

|vyo = gt |

|0 = vyo2 + 2gdy |

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|no dy |

|no t |

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|solve for unknown in the x direction with dx = vxt |

|data chart—landing point |

|direction |

|d |

|vo |

|vt |

|a |

|t |

| |

|y |

|0 |

|vyo |

|-vyo |

|-10 m/s2 |

|t |

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|x |

|dx |

|vx |

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|solve for unknown in the vertical direction with |

|2vyo = gt |

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|no dy |

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|solve for unknown in the x direction with dx = vxt |

d. projectile launched at angle θo from height h

|determine vxo = vocosθ and vyo = vosinθ |

|data chart—landing point |

|direction |

|d |

|vo |

|vt |

|a |

|t |

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|y |

|dy |

|vyo |

|vyt |

|-10 m/s2 |

|t |

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|x |

|dx |

|vx |

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| |

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|solve for unknown in the vertical direction with |

|dy = vyot + ½at2 |

|vyt = vyo + at |

|vyt2 = vyo2 + 2ady |

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|no vyt |

|no dy |

|no t |

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|solve for unknown in the x direction with dx = vxt |

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A. Vectors

Questions 1-12 Briefly explain your answer.

1. If two vectors A + B = 0, what can you say about the magnitude and direction of the two vectors?

(A) same magnitude but opposite direction

(B) same magnitude and direction

(C) different magnitude and opposite direction

(D) different magnitude and same direction

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2. Given vectors A + B = C and |A|2 + |B|2 = |C|2, how are the vectors A and B oriented with respect to each other?

(A) perpendicular

(B) parallel in the opposite direction

(C) parallel in the same direction

(D) 45o from each other

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3. Given vectors A + B = C and |A| + |B| = |C|, how are the vectors A and B oriented with respect to each other?

(A) perpendicular

(B) parallel in the opposite direction

(C) parallel in the same direction

(D) 45o from each other

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4. If each component of a vector is doubled, what happens to the angle of the vector?

(A) doubles (B) the same (C) reduced by half

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5. If each component of a vector is equal in length, what is the angle of the vector?

(A) 30o (B) 45o (C) 60o (D) 30o or 60o

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6. If one of the components is half the length of the vector, what is the angle of the vector?

(A) 30o (B) 45o (C) 60o (D) 30o or 60o

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Questions 7-8 Vx = 30 units Vy = 40 units.

7. What is the magnitude of the resultant V?

(A) 30 units (B) 40 units (C) 50 units (D) 70 units

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8. What is the angle of the resultant V?

(A) 37o (B) 45o (C) 53o (D) 37o or 53o

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9. You are adding vectors of length 20 units and 40 units. What is the only possible resultant magnitude?

(A) 0 (B) 18 (C) 37 (D) 64

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Questions 10-11 Two students are tossing a ball back and forth in the aisle of a train. The velocity of the ball is vB and the velocity of the train is vT.

10. What is the velocity of the ball with respect the ground if the velocities of the ball and train are in the same direction?

(A) vB + vT (B) vB – vT (C) 0

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11. What is the velocity of the ball with respect the ground if the velocities of the ball and train are in the opposite direction?

(A) vB + vT (B) vB – vT (C) 0

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12. Alice and Bill set off across a river at the same time and with the same speed with respect to the water. Alice heads straight across and is pulled downstream by the current. Bill heads upstream at an angle so as to arrive at a point opposite the starting point. Who reaches the opposite side first?

(A) Alice (B) Bill (C) tie

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13. A student jogs 100 m at 60o north of east. What are the x- and y-components of the displacement vector?

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14. Determine the resultant for each situation below.

a. 8 m north + 6 m north

| |magnitude |direction |

|8 m ↑ + 6 m ↑ | | |

|8 m ↑ + 6 m ↓ | | |

|8 m ↑ + 6 m ← | | |

15. a. Label the x-y grid below:

(1) north, south, east, west

(2) 0o, 90o, 180o, and 270o

b. Draw a displacement vector of 20 m 30o north of east

(1 cm = 10 m scale)

16. Adding Vectors

a. Measure the magnitude and angle of vector A and B

B

A

|Vector |Magnitude (L) |Angle (θ) |

|A | | |

|B | | |

b. Calculate the x- and y-components of the two vectors and then add the x-components to determine the resultant

x-component and repeat with the y-components.

|Vector |x-component |y-component |

|A | | |

|B | | |

|R | | |

c. Calculate the magnitude and angle of the resultant.

|Vector |magnitude |direction |

|R | | |

d. Draw the resultant vector on the diagram and measure its magnitude and angle.

|Vector |Magnitude |Angle |

|R | | |

e. Determine the percent difference between the two.

|Magnitude |Angle |

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17. Consider the following vectors, A and B.

|Vector |A |B |

|Magnitude |8.0 cm |4.0 cm |

|Angle |25o |150o |

a. Calculate the x-component and y-component of each vector and then determine the x-component and y-component of the resultant R.

|Vector |x-component |y-component |

|A | | |

|B | | |

|R | | |

b. Calculate the magnitude and angle of the resultant.

|Vector |Magnitude |Angle |

|R | | |

18. Consider the following vectors, A, B and C.

|Vector |Magnitude |Angle |

|A |12 cm |45o |

|B |8 cm |135o |

|C |10 cm |-75o |

a. Calculate the x- and y-component of vectors A, B and C and then determine the resultant's components.

|Vector |x-component |y-component |

|A | | |

|B | | |

|C | | |

|R | | |

b. Calculate the magnitude and angle of the resultant.

|Vector |magnitude |direction |

|R | | |

19. A boat travels at 1.85 m/s (vboat) in a river whose current (vwater) is 1.20 m/s west. Determine the velocity of the boat with respect to the ground for each situation.

a. the boat is headed due east?

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b. the boat is headed due north?

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20. Alice's boat heads due east at 4 m/s across a river that is 800 m across. The water current of 3 m/s south pushes the boat downstream.

a. What is the velocity of the boat with respect to Earth?

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b. What is the boat's heading?

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c. How far did the boat travel?

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d. How much time did it take to cross the river?

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21. Bill's boat has a maximum speed is 4 m/s. He directs the boat upstream to correct for a 3 m/s current going south, so that he goes the shortest distance due east to cross the 800 m river.

a. What is the velocity of the boat with respect to Earth?

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b. What is the boat's heading?

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c. How much time did it take to cross the river?

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22. A FedEx plane travels from New York to Los Angeles (4,000 km), stays in LA for 90 minutes to refuel and unload and reload cargo, and then returns to NY. The plane's air speed is 800 km/hr and the west to east wind current is 100 km/hr. How long does it take for the round trip?

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B. Projectile Motion

Questions 23-36 Briefly explain your answer.

23. A small cart is rolling a constant velocity on a flat track. It fires a ball straight up into the air as it moves. Where does the ball land?

(A) In front of the cart

(B) in the cart

(C) behind the cart

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24. A small cart accelerates on a flat track. It fires a ball straight up into the air as it speeds up. Where does the ball land?

(A) In front of the cart

(B) in the cart

(C) behind the cart

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25. A small cart accelerates down an inclined track. the ball is fired straight out (perpendicular to the track) as the cart accelerates down the track. Where does the ball land?

(A) In front of the cart

(B) in the cart

(C) behind the cart

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26. You drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will

(A) lag behind the plane as it falls

(B) remain directly beneath the plane as it falls

(C) move ahead of the plane as it falls

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Questions 27-28 From the same height and at the same time, ball A is dropped and ball B is fired horizontally.

27. Which ball will land first?

(A) A (B) B (C) tie

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28. Which ball has the greater velocity when it lands?

(A) A (B) B (C) tie

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29. A projectile is launched at an angle of 30o. At what point in its trajectory does the projectile have the lowest speed?

(A) at the start (B) at the middle (C) at the end

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30. Which punt has the longest hang time, or do they tie?

A B C

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31. Which throw from the outfield will reach second base first?

A B second base

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32. The two identical projectiles use the same launcher. Path A takes place on Earth and path B on the Moon, where g is 1/6 Earth's.

A B

Which accounts for the greatest difference between the two paths?

(A) The initial velocity on the Moon is greater than on Earth.

(B) The downward acceleration on the Moon is less than on Earth.

(C) The airless environment on the Moon allows for the horizontal velocity to remain constant.

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33. The spring loaded gun can launch rubber balls from ground level at different angles with the same launch speed. What is the best angle for maximizing the horizontal distance that the ball travels before landing? (Assume no air resistance)

(A) 0o (B) 30o (C) 45o (D) 60o

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Questions 34-36 You are at ground level and aim your spring loaded dart gun directly at your friend.

34. Your friend is standing on a 3 foot wall and jumps off of the wall just as you shoot the gun. Does the dart hit him?

(A) yes (B) no

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35. Your friend is standing on an 8 foot wall and jumps off of the wall just as you shoot the gun. Does the dart hit him?

(A) yes (B) no

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36. Your friend is standing on an 8 foot wall aiming his dart gun directly at you. At the instant he shoots his gun you shoot yours. Do the darts hit each other?

(A) yes (B) no

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37. A ball is thrown horizontally at 10 m/s from a bridge that is 20 m above the water.

a. Complete the chart

| |d |vo |vt |a |t |

|y | | | | | |

|x | | | | |

b. How long does it take for the ball to hit the water?

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c. How far horizontally does the ball travel?

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38. A rescue plane (traveling horizontally at 70 m/s) wants to drop supplies to climbers on a rocky ridge 235 m below.

a. Complete the chart

| |d |vo |vt |a |t |

|y | | | | | |

|x | | | | |

b. How long does it take for the supplies to fall 235 m?

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c. How far in advance must the supplies be dropped?

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d. What is the line of sight angle from the plane to ridge?

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e. What is the vertical velocity, vyt?

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f. With what speed do the supplies land?

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39. A ball is thrown from the top of a 20 m building with an initial velocity of 10 m/s at various angles. Complete the charts.

a. The ball is thrown at 0o

| |d |vo |vt |a |t |

|y | | | | | |

|x | | | | |

|t | |

|dx | |

|vyt| |

|vy | |

b. The ball is thrown at 90o

| |d |vo |vt |a |t |

|y | | | | | |

|x | | | | |

|t | |

|vyt| |

c. The ball is thrown at -90o

| |d |vo |vt |a |t |

|y | | | | | |

|x | | | | |

|t | |

|vyt| |

d. The ball is thrown at 37o

| |d |vo |vt |a |t |

|y | | | | | |

|x | | | | |

|t | |

|dx | |

|vyt| |

|vy | |

40. A soccer ball is kicked 20 m/s at an angle of 30o.

a. Determine the x- and y- components of the velocity vo?

|vxo | |

|vyo | |

b. Complete the chart for the ball when it lands.

| |d |vo |vt |a |t |

|y | | | | | |

|x | | | | |

c. What is the total time in the air?

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d. What is the total distance the ball travels horizontally?

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e. Complete the chart for the ball at its highest point.

| |d |vo |vt |a |t |

|y | | | | | |

|x |

g. Graph the following.

| |Displacement |Velocity |Acceleration |

|Horizontal | |t | |t | |t |

| | | | | | | |

| | | | | | | |

|Vertical | |t | | |t | |t |

| | | | | | | | |

| | | | | | | | |

41. A projectile, fired with velocity vo and angle θ, lands on a cliff, which is 195 m away and 155 m high, in 7.6 s.

a. Complete the chart

| |d |vo |vt |a |t |

|y | | | | | |

|x | | | | |

b. What is vxo?

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c. What is vyo?

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d. What is vo?

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e. What is θ?

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Catapult Lab

42. Determine the optimum setting (listed below) for the ping-pong ball catapult by measuring the distance in the air.

L1: length of arm from rubber band to axel

L2: length of arm from cup to axel

θ: launch angle

A: rubber band anchor

a. Complete the data chart

|L1 |L2 |θ |A |Distance |

| | | | |1 |2 |3 |Avg |

|1 |4 |8 |11 | | | | |

|2 |4 |8 |11 | | | | |

|3 |4 |8 |11 | | | | |

|Max |5 |8 |11 | | | | |

| |6 |8 |11 | | | | |

| |7 |8 |

|vxo | | |

|vyo | | |

|vo | | |

|θ | | |

43. A ball is thrown horizontally at 20 m/s from a bridge that is 80 m above the water.

| |d |vo |vt |a |t |

|y | | | | | |

|x | | | | |

a. How long does it take for the ball to hit the water?

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b. How far horizontally does the ball travel?

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44. A rescue plane, traveling horizontally at 100 m/s, wants to drop supplies to climbers on a rocky ridge 125 m below.

| |d |vo |vt |a |t |

|y | | | | | |

|x | | | | |

a. How long does it take for the supplies to fall 125 m?

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b. How far in advance must the supplies be dropped?

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c. What is the line of sight angle from the plane to ridge?

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d. What is the vertical velocity vyt?

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e. With what speed do the supplies land?

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45. A soccer ball is kicked 30 m/s at an angle of 60o.

a. Determine the x- and y- components of the velocity vo.

|vxo | |

|vyo | |

b. Complete the chart for the ball when it lands.

| |d |vo |vt |a |t |

|y | | | | | |

|x | | | | |

c. What is the total time in the air?

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d. What is the total distance the ball travels horizontally?

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e. What is the maximum height?

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46. A student throws a ball that lands on a cliff that is 60 m away and 20 m above the student. The ball is in the air for 4 s.

| |d |vo |vt |a |t |

|y | | | | | |

|x | | | | |

Determine the following for the ball's initial velocity

a. horizontal component

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b. vertical component

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c. magnitude

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d. direction

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Practice Multiple Choice (no calculator)

Briefly explain why the answer is correct in the space provided.

|1 |

2. Two spheres, A and B, are thrown horizontally from the top of a tower; vA = 40 m/s and vB = 20 m/s. Which is true of the time T in the air and horizontal distance d traveled?

(A) TA = TB, dA = dB (B) TA = TB, dA = 2dB

(C) TA = TB, 2dA = dB (D) TA > TB, dA > dB

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3. A plane flying horizontally at 100 m/s drops a crate. What is the horizontal component of the crate's velocity just before it strikes the ground 3 seconds later?

(A) 0 m/s (B) 100 m/s (C) 300 m/s (D) 400 m/s

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Questions 4-5 A ball slides off a roof, h = 10 m, with velocity,

vx = 5 m/s.

vx

h

4. How much time is the ball in the air?

(A) 0.5 s (B) 0.7 s (C) 1.4 s (D) 2 s

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5. How far does the ball travel horizontally before hitting the ground?

(A) 2.5 m (B) 5 m (C) 7 m (D) 10 m

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6. A projectile is fired with initial velocity, vo, at an angle θo.

[pic]

Which pair of graphs represent the vertical components of the velocity vy and acceleration ay of the projectile as functions of time t?

(A) vy ay (B) vy ay

t t t t

(C) vy ay (D) vy ay

t t t t

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Questions 7-10 A projectile is fired from the ground with an initial velocity of 250 m/s at an angle of 37o above horizontal.

7. What is the vertical component of the initial velocity?

(A) 100 m/s (B) 150 m/s (C) 200 m/s (D) 250 m/s

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8. How long is the projectile in the air (assume the projectile lands at the same elevation that it was fired)?

(A) 25 s (B) 30 s (C) 45 s (D) 50 s

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9. What is the horizontal component of the initial velocity?

(A) 100 m/s (B) 150 m/s (C) 200 m/s (D) 250 m/s

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10. How far did the projectile travel horizontally before it struck the ground?

(A) 6,000 m (B) 7,000 m (C) 9,000 m (D) 10,000 m

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11. A projectile is fired with an initial velocity of 100 m/s at an angle θ above the horizontal. If the projectile's initial horizontal speed is 60 m/s, then angle θ measures approximately

(A) 30o (B) 37o (C) 40o (D) 53o

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12. A rock is dropped from the top of a 45-m tower, and at the same time a ball is thrown from the top of the tower in a horizontal direction. Air resistance is negligible. The ball and the rock hit the level ground a distance of 30 m apart. The horizontal velocity of the ball thrown was most nearly

(A) 5 m/s (B) 10 m/s (C) 15 m/s (D) 20 m/s

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13. A projectile is fired from the surface of the Earth with a speed of 200 m/s at an angle of 30° above the horizontal. If the ground is level, what is the maximum height reached by the projectile?

(A) 5 m (B) 10 m (C) 500 m (D) 1,000 m

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14. Vectors V1 and V2 have equal magnitudes and represent the velocities of an object at times t1 and t2, respectively.

[pic]

The direction of acceleration between time t1 and t2 is

(A) √ (B) © (C) ∇ (D) ∏

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Questions 15-16 A person on the west side of a river, which runs north to south, wishes to cross the 1-km wide river to a point directly across from his starting point. The river current is 3 km/hr and the boat's maximum speed is 5 km/hr.

15. What direction should the person direct the boat?

(A) due east (B) due north

(C) 37o north of east (D) 53o north of east

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16. How long will it take to reach the dock?

(A) 1/5 hr (B) 1/4 hr (C) 1/2 hr (D) 1 hr

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17. A stationary observer on the ground sees a package falling with speed v1 at an angle to the vertical. At the same time, a pilot flying horizontally at constant speed relative to the ground sees the package fall vertically with speed v2. What is the speed of the pilot relative to the ground?

(A) v1 + v2 (B) v1 – v2

(C) v2 – v1 (D) (v12 – v22)½

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18. How much time is a rock in the air if it is thrown horizontally off a building from a height h with a speed vo?

(A) (hvo)½ (B) h/vo (C) hvo/g (D) (2h/g)½

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19. A spring-loaded gun can fire a projectile to a height h if it is fired straight up. If the same gun is pointed at an angle of 45° from horizontal, what maximum height can now be reached by the projectile?

(A) h/4 (B) h/2√2 (C) h/2 (D) h/√2

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Practice Free Response

1. A physics student kicks a ball across the football field. It travels 50 m in 3.2 seconds.

a. What is the x-component of the initial velocity?

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b. What is the y-component of the initial velocity?

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c. What are the initial velocity's magnitude and direction?

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d. Graph the following.

dx vx ax

t t t

dy vy ay

t t t

2. A rescue plane, traveling horizontally at 70 m/s, drops supplies to mountain climbers on a rock ridge 235 m below.

[pic]

a. How far in advance, x, are the supplies dropped?

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b. Suppose, instead, that the plane releases the supplies a horizontal distance of 425 m in advance of the climbers. What vertical velocity (up or down) is given to the supplies so that they reach the climbers?

[pic]

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3. Is it possible to hit a horizontal serve (without spin) that falls within the service box and clear the 0.90 m net?

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