AP Physics 2: Kinematics—Two Dimension Name __________________________A. Vectors

1. vectors and scalarsa. 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 direction2. magnitude arrow's length

2. addition of vectors—tail to tip methodvectors 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 is

measured counterclockwise from +x-axis (0o) y = 90o

B B A A

x = 0o

x-component and y-component for each vector are calculated (R = magnitude; = direction)

cos = adjacent/hypotenuse Rx = Rcossin = opposite/hypotenuse Ry = Rsin

y Rx = Ax + Bx Bx= BcosB

B By = BsinB

Ry = Ay + By R Ay = AsinA

A x Ax = AcosA

Rx = Ax + Bx = AcosA + BcosB

Ry = Ay + By = AsinA + BsinB R = (Rx

2 + Ry2)½

tan = Ry/Rx = tan-1(Ry/Rx) add 180o to when Rx is negative4. 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

B. Projectile Motion1. projectile's horizontal velocity is constant 2. projectile's vertical velocity is immediately affected by

the downward acceleration of gravity3. solving projectile motion problems

a. general solution determine vxo = vocos and vyo = vosin 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 directionsdirection d vo vt a t

y dy vyo vyt -10 m/s2

tx dx vx

solve for unknown in the vertical direction with dy = vyot + ½gt2 vyt = vyo + gt vyt

2 = vyo2 + 2gdy

no vyt no dy no t solve for unknown in the x direction with dx = vxt

b. projectile launched horizontally from height h data chartdirection d vo vt a t

y -h 0 vyt -10 m/s2

tx dx vx = vo

solve for unknown in the vertical direction with h = ½gt2 vyt = gt vyt

2 = 2gh solve for unknown in the x direction with dx = vxt

c. projectile launched at angle o from height 0

determine vxo = vocoso and vyo = vosino

data chart—highest pointdirection d vo vt a t

y dy vyo 0 -10 m/s2

tx dx vx

solve for unknown in the vertical direction with vyo = gt 0 = vyo

2 + 2gdy

no dy no t solve for unknown in the x direction with dx = vxt data chart—landing pointdirection d vo vt a t

y 0 vyo -vyo -10 m/s2

tx dx vx

solve for unknown in the vertical direction with

2vyo = gtno dy

solve for unknown in the x direction with dx = vxtd. projectile launched at angle o from height h

determine vxo = vocos and vyo = vosin data chart—landing pointdirection d vo vt a t

y dy vyo vyt -10 m/s2

tx dx vx

solve for unknown in the vertical direction with dy = vyot + ½at2 vyt = vyo + at vyt

2 = vyo2 + 2ady

no vyt no dy no t solve for unknown in the x direction with dx = vxt

A. VectorsQuestions 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

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

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

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

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

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

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

8. What is the angle of the resultant V?(A) 37o (B) 45o (C) 53o (D) 37o or 53o

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

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

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

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

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

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

B A

Vector Magnitude (L) Angle ()AB

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

ABR

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 AngleR

e. Determine the percent difference between the two. Magnitude Angle

17. Consider the following vectors, A and B.Vector A BMagnitude

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?

b. the boat is headed due north?

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?

b. What is the boat's heading?

c. How far did the boat travel?

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

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?

b. What is the boat's heading?

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

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?

B. Projectile MotionQuestions 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

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

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

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

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

28. Which ball has the greater velocity when it lands?(A) A (B) B (C) tie

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

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

A B C

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

A B second base

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.

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

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

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

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

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 tyx

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

c. How far horizontally does the ball travel?

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 tyx

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

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

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

e. What is the vertical velocity, vyt?

f. With what speed do the supplies land?

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 tyx

t

dx

vyt

vy

b. The ball is thrown at 90o

d vo vt a tyx

t

vyt

c. The ball is thrown at -90o

d vo vt a t

yx

t

vyt

d. The ball is thrown at 37 o d vo vt a t

yx

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

yx

c. What is the total time in the air?

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

e. Complete the chart for the ball at its highest point.d vo vt a t

yxf. What is the maximum height?

g. Graph the following.Displacement Velocity Acceleration

Horizontal t t t

Vertical t t t

41. A projectile, fired with velocity vo and angle , 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 tyxb. What is vxo?

c. What is vyo?

d. What is vo?

e. What is ?

Catapult Lab42. 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 axelL2: length of arm from cup to axel: launch angleA: rubber band anchor

a. Complete the data chartL1 L2 A Distance

1 2 3 Avg1 4 8 112 4 8 113 4 8 11

Max

5 8 116 8 117 8 11

max

9 1110 11

max1213

Adjust the catapult to the maximum settings. Launch the ping-pong ball and measure horizontal distance in the air, time in the air, and launch arm height.b. Collect the following data.

L1 L2 A dx t dy

c. Calculate the following from the data.Formula Calculation

vxo

vyo

vo

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

d vo vt a tyxa. How long does it take for the ball to hit the water?

b. How far horizontally does the ball travel?

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 tyxa. How long does it take for the supplies to fall

125 m?

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

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

d. What is the vertical velocity vyt?

e. With what speed do the supplies land?

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

yx

c. What is the total time in the air?

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

e. What is the maximum height?

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 tyxDetermine the following for the ball's initial velocitya. horizontal component

b. vertical component

c. magnitude

d. direction

Practice Multiple Choice (no calculator)Briefly explain why the answer is correct in the space provided.1 2 3 4 5 6 7 8 9 10C B B C C D B B C A11 12 13 14 15 16 17 18 19D B C D C B D D C

1. A machine launches a tennis ball at an angle = 25° above the horizontal at an initial velocity vo = 14 m/s. Which combination of changes must produce an increase in time of flight of a second launch?(A) decrease and vo (B) decrease and increase vo

(C) increase and vo (D) increase and decrease vo

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

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

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

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

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

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

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

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

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

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

11. A projectile is fired with an initial velocity of 100 m/s at an angle above the horizontal. If the projectile's initial horizontal speed is 60 m/s, then angle measures approximately (A) 30o (B) 37o (C) 40o (D) 53o

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

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

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

The direction of acceleration between time t1 and t2 is(A) (B) (C) (D)

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

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

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 – v2

2)½

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

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/22 (C) h/2 (D) h/2

Practice Free Response1. 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?

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

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

d. Graph the following.dx vx ax

t t tdy 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.

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

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?

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