Physics



Physics 1: Kinematics—One Dimension Review Name __________________________

A. Measurement

1. science knowledge is advanced by observing patterns (laws) and constructing explanations (theories), which are supported by repeatable experimental evidence

a. theory lasts until disproven

b. theory is never 100 % certain

2. uncertainty in measurements

a. precision and accuracy

1. precise = consistent (even if incorrect)

2. accurate = correct (even if inconsistent)

b. data analysis

1. accuracy is measured by percent difference

% Δ = 100|mean – true|/true

2. precision is measured by percent deviation

% Δ = 100Σ|trial – mean|/N(mean)

(N is number of trials)

c. significant figures (sf) indicate level of certainty

[pic]

measurement includes all certain (numbered) plus one estimated value ∴ 7.5 cm (2 sf)

d. rules for counting significant figures

1. all nonzero digits are significant

2. zero is sometimes significant, sometimes not

a. example: 0.00053000021000

never always ?

b. (?) decimal vs. no decimal

1. significant with decimal: 120. (3 sf)

2. not significant w/o decimal: 120 (2 sf)

3. exact numbers (metric conversions, counting or written numbers) have infinite number of sf

e. rules for rounding off calculations

1. limited by least accurate measurement

2. x, ÷: answer has the same number of sf as the measurement with the fewest

3. +, –: answer has same end decimal position as measurement with left most end position

3. SI measuring system

a. summary chart

|Measurement |SI standards |

|mass |kilogram (kg) |

|length |meter (m) |

|area |square meter (m2) |

|volume |cubic meter (m3) |

|temperature |kelvin (K) |

|time |second (s) |

b. prefixes system (x 10X)

1. G9, M6, k3, c-2, m-3, µ-6, n-9, p-12

a. km → m: 8.75 km = 8.75 x 103 m

b. m → km:

455 m x 1 km/103 m = 0.455 km

2. squared/cubed prefix:

1 cm2 = 1 x (10-2)2 m2

1 cm3 = 1 x (10-2)3 m3

3. 1 mL = 1 cm3

4. dimensional analysis math technique

455 kg x 103 g x (10-2)3 m3 = 0.455 g

m3 1 kg cm3 cm3

4. scientific notation: C x 10n

a. conversion from decimal to scientific notation

1. 1,200,000 = 1.2 x 106

2. 0.0000012 = 1.2 x 10-6

b. significant figures

1. C contains only significant figures

2. 1200 with 3 significant figures = 1.20 x 103

B. Data Analysis Using Graphs

1. graphing data (i.e. position "x" vs. time "t")

a. Cartesian axis

1. x-axis is independent variable (t)

2. y-axis is the dependent variable (x)

b. axis labels

1. measurement and units, i.e. position (m)

2. spread out scale to fit entire graph using the origin as zero (unless told otherwise) and using equal, logical increments

c. coordinate subscripts

1. subscript o (xo) indicate starting position (usually the origin: xo = 0)

2. subscript t (xt) indicate position at time “t” (often the t is dropped: xt = x)

d. + vs. –: direction can be ahead (+) or behind (–)

2. graphing 2-dimentional position

a. north-south-east-west directions

a. x-axis is east-west (+ is east)

b. y-axis is north-south (+ is north)

b. vertical-horizontal directions

a. x-axis is horizontal (+ is away)

b. y-axis is vertical (+ is up)

c. polar coordinate system (r, θ)

1. r is distance from the origin

2. θ is angle measured Θ from +x

3. translation to Cartesian coordinates

a. x = rcosθ

b. y = rsinθ

3. best fit line (i.e. position "x" vs. time "t")

a. individual coordinates may not line up exactly because of small experimental errors

b. best fit line shows data trend

1. spacing between data points and best fit line are equal above and below best fit line

2. averages out errors

3. data that is well outside best fit line should be repeated

c. interpreting graphs

|y = k |y = kx |y = -kx |y = kx2 |y = k/x |

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d. values can be determined from the graph

1. speed (velocity) = Δposition/Δtime: (v = Δx/Δt)

2. graph position (y-axis) vs. time (x-axis) and slope = speed (v)

3. graph speed (y-axis) vs. time (x-axis) and area between the graph and x-axis = change in position (Δx)

4. graph sequence

| |slope → |slope → | |

| |← area |← area | |

|y = x |y' = k |y" = 0 |

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|y = x2 |y' = x |y" = k |

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5. slope of a curve = slope of tangent line

tangent line

|C. Kinematics |

|1. displacement (distance): Δx = d = x – xo (m) |

|2. change in time: Δt = t – to (s) (usually to = 0 ∴ Δt = t) |

|3. velocity (speed): vav = d/t (m/s) |

|4. constant motion vs. acceleration |

|a. when an object is left alone it continues at a constant speed in a straight|

|line (inertia) |

|b. when an object is pushed/pulled it speeds up or slows down or changes |

|direction (acceleration) |

|5. acceleration: a = (vt – vo)/t (m/s2) |

|a. instantaneous velocity, vt, is velocity at time, t |

|special case: if vo = 0, then vt = 2vav |

|b. falling objects |

|1. At a given location and in the absence of air resistance, all objects fall |

|with the same constant acceleration. (Galileo) |

|2. acceleration due to gravity, g, at sea level is about 9.80 m/s2 (use 10 |

|m/s2 in calculations except for labs) |

|6. positive and negative case |

|a. direction can be "forward (+) or backward (–) |

|b. velocity and displacement always have the same sign |

|c. acceleration can have different sign |

|1. when velocity and acceleration have the same sign: speeds up |

|2. when velocity and acceleration have different signs: slows down |

|7. solving kinematics problems |

|a. constant motion (a = 0) |

|draw diagram |

|complete chart with two numbers and one letter |

|d |

|vav |

|t |

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|use the definition of average velocity: vav = d/t |

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|b. accelerated motion (a ≠ 0) |

|draw diagram |

|complete chart with given information (three numbers and two letters) |

|d |

|vo |

|vt |

|a |

|t |

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|use the formula that contains numbers + letter of unknown, but is missing |

|unused letter |

|Unused Letter |

|Formula |

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|a |

|d = ½(vo + vt)t |

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|d |

|vt = vo + at |

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|vt |

|d = vot + ½at2 |

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|t |

|vt2 = vo2 + 2ad |

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|8. deriving the kinematic formulas (there are nearly 100 formulas in physics |

|and it is easy to forget or confuse them, so when possible learn how they are |

|derived) |

|Steps |

|Algebra |

| |

|start with definitions of vav |

|solve for d |

|vav = ½(vt + vo) = d/t |

|d = ½(vo + vt)t |

| |

|start with definition of a |

|solve for vt |

|a = (vt – vo)/t |

|vt = vo + at |

| |

|start with formula # 1 |

|substitute vt from formula #2 |

|simplify |

|d = ½(vo + vt)t |

|d = ½(vo + vo + at)t |

|d = vot + ½at2 |

|(area under v vs. t graph) |

| |

|start with formula #1 |

|solve for t |

|start with formula #2 |

|solve for t |

|set equal to each other |

|cross multiple |

|difference of perfect squares |

|rearrange |

|d = ½(vo + vt)t |

|t = 2d/(vo + vt) |

|vt = vo + at |

|t = (vt – vo)/a |

|2d/(vo + vt) = (vt – vo)/a |

|2ad = (vt + vo)(vt – vo) |

|2ad = vt2 – vo2 |

|vt2 = v02 + 2ad |

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Practice Problems

Kinematics (skipping 1-4)

5. You drive for 30 minutes at 30 mph and then for another 30 minutes at 50 mph. What is your average speed?

(A) < 40 mph (B) 40 mph (C) > 40 mph

6. You drive for 30 miles at 30 mph and then 30 miles at 50 mph. What is your average speed for the whole trip?

(A) < 40 mph (B) 40 mph (C) > 40 mph

7. Your instantaneous velocity is zero. What must also be zero?

(A) displacement (B) acceleration

(C) Both (D) neither

8. Your acceleration is zero. What must also be zero?

(A) displacement (B) velocity

(C) Both (D) neither

9. Your car has negative velocity and positive acceleration, it is

(A) speeding up while going backward

(B) speeding up while going forward

(C) slowing down while going backward

(D) slowing down while going forward

10. A thrown ball on its upward trajectory has

(A) positive velocity and acceleration

(B) positive velocity and negative acceleration

(C) negative velocity and positive acceleration

(D) negative velocity and acceleration

11. You throw a ball straight up into the air; it reaches a maximum height, and then returns to your hand. At what point in its flight is the acceleration maximum?

(A) just after it leaves your hand

(B) at the top of its flight

(C) just before it returns to your hand

(D) it is constant for the entire flight

12. When throwing a ball straight up, which is true about velocity, v, and its acceleration, a, at the highest point?

(A) v = 0, a = 0 (B) v = 0, a ≠ 0 (C) v ≠ 0, a = 0

13. Alice and Bill are at the top of a building. Alice throws her ball upward. Bill simply drops his ball. Which ball has the greater acceleration just after being released?

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

14. Alice and Bill are at the top of a building and throw balls with equal speed, but Alice throws her straight down, while Bill throws his straight up. Which ball hits the ground with the greater speed (assume there is no air resistance)?

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

15. You throw a ball straight up in the air at 10 m/s. What is the ball's speed when it returns to your hand? (Assume no air resistance)

(A) < 10 m/s (B) 10 m/s (C) > 10 m/s

Questions 16-17 You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock.

16. What happens to their separation as they continue to fall?

(A) decreases (B) same (C) increases

17. What happens to the difference in velocity between the two?

(A) decreases (B) constant (C) increases

18. A person walks 7 m east in 7 s then walks 3 m west in 3 s.

a. What is the total distance that the person walked?

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b. What is the displacement?

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c. What is the person's average velocity?

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d. What is the person's average speed?

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19. A person runs from 50 m to 30 m in 2 s.

a. What is the runner's displacement?

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b. What is the runner's velocity?

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20. A runner travels 150 m in 17 s.

a. Fill in the constant velocity chart with the data. Fill the missing box with the letter of the variable.

|d |vav |t |

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b. Determine the runner's average speed.

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21. How far can a cyclist travel in 2 hours at 5 m/s?

|d |vav |t |

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22. Determine the acceleration for the following situations.

a. A car initially at rest is traveling 15 m/s 5 s later.

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b. A car's velocity is 15 m/s at t = 0 s and 5 m/s at t = 5 s.

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23. A car is traveling at 15 m/s and comes to a stop in 3 s.

a. Fill in the acceleration chart (use letters for unknowns).

|d |vo |vt |a |t |

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b. Calculate the acceleration using the kinematic formula that excludes the letter that is NOT the unknown.

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24. A plane accelerates from rest at 2 m/s2 and reaches a final velocity of 28 m/s before taking off.

|d |vo |vt |a |t |

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Determine the minimum length of runway.

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25. A rock is dropped from a 100 m cliff.

|d |vo |vt |a |t |

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Determine the time that the rock is in the air.

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26. When the space shuttle is launched, it reaches a velocity of 900 m/s in 3 minutes.

|d |vo |vt |a |t |

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Determine how far the space shuttle travels in the 3 minutes.

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27. A rock, dropped from a bridge, takes 5 s to hit the water.

| d |vo |vt |a |t |

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How high is the cliff?

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28. A car accelerates from 10 m/s to 30 m/s in 10 s.

|d |vo |vt |a |t |

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a. What is the car's acceleration?

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b. How far did the car travel during acceleration?

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29. A driver in a car traveling at 30 m/s sees a deer in the road. It takes 0.5 s before he reacts and steps on the breaks,

|d |v |t |

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a. How far does the car travel during the 0.5 s?

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The breaks can decelerate the car at -6.0 m/s2.

|d |vo |vt |a |t |

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b. How far does the car travel during deceleration?

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c. What is the total distance traveled?

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30. Bill, at a stop sign, sees Alice drive by at a constant velocity of 20 m/s. Bill accelerates to catch up to Alice.

a. What is Bill's top speed when he reaches Alice if he averaged Alice's speed while catching up.

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Bill accelerates from rest at 2.5 m/s2.

|d |vo |vt |a |t |

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b. How long does it take Bill to catch up to Alice?

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c. How far does Bill travel while catching up to Alice?

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31. A ball is thrown upward with vo = +20 m/s.

a. What is the acceleration (including sign) due to gravity?

|on the way up |at the ball's highest |on the way down |

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b. Complete the list of variables for the highest point.

|d |vo |vt |a |t |

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(1) How high does the ball rise?

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(2) How much time does it take to reach the highest point?

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c. Consider the symmetry of the ball's flight.

|How much time is the ball in the air? | |

|How fast is the ball when it returns? | |

Graphing One Dimensional Motion

Questions 32-34 Consider the graph of position vs. time for cars A and B.

x A

B

0 1 2 3 4 5 6 7 8 9 t

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

32. Which car is accelerating?

33. Which car has greater displacement from 0 to 7 s?

34. Which car has greater velocity at 7 s?

Questions 35-36 Consider the velocity vs. time graphs below.

v A

0 t

B

C

35. Which represent a dropped ball right after it leaves you hand but before it hits the floor?

36. Which represent a ball that is thrown straight up in the air and falls back into its original height?

37. A very bouncy ball is dropped and it hits the floor and returns to the original height. Draw a graph of the velocity vs. time for this event.

v

0 t

38. The vertical displacement (d) of an elevator as a function of time (t) is shown below.

d (m)

[pic]

t (s)

a. Calculate the velocity for each time interval.

|0 s to 8 s |10 s to 18 s |20 s to 24 s |

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b. Calculate the acceleration for each time interval.

|8 s to 10 s |18 s to 20 s |

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c. Graph the velocity as a function of time.

v (m/s)

|2 | | | | | | | |

| |t (s) |

39. The graph of velocity versus time for a cart is given below.

v (m/s)

|4 | | | | | | | |

| |t (s) |

a. At what times was the cart at rest?

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b. At what time does the cart return to its original position?

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c. Determine the acceleration for each time interval.

|0 to 2 s |2 s to 3 s |

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|3 s to 5 s |5 s to 7 s |

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|7 s to 11 s |11 s to 12 s |

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d. Graph the acceleration of the cart as a function of time.

a (m/s2)

|1 | | | | | | | |

| |t (s) |

e. Graph the displacement of the cart as a function of time.

d (m)

|6 | | | | | | | |

| |t (s) |

40. A student tries to minimize the time it takes to go between two stop lights without speeding. He accelerates at 2.5 m/s2 until he reaches the speed limit of 20 m/s.

a. How much time does it take accelerate to 20 m/s?

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b. How far does he travel during the acceleration?

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At the next stop light he decelerates to a stop at 4 m/s2.

c. How much time does this take?

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d. How far does he travel during deceleration?

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e. The stop lights are 200 m apart. How much time does it take to travel the middle distance at constant speed?

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f. What is the total time that it takes the driver to go from stop light to stop light if his initial reaction time is 0.3 s?

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41. A plane must land on a 605 m long run way. What is the plane acceleration if its landing speed is 90 m/s?

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42. Bill, traveling at 30 m/s, is passed by Alice traveling at a constant 40 m/s. Bill accelerates at 0.8 m/s2 to catch up.

a. How fast is Bill going when he catches Alice?

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b. How long does it take Bill to catch up to Alice?

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c. How far does Bill travel while catching up to Alice?

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43. A ball is thrown upward with vo = 40 m/s.

a. What is the direction of acceleration due to gravity?

|on the way up |at the highest point |on the way down |

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b. How high does the ball rise?

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c. How much time does it take to reach the highest point?

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

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e. How fast is the ball traveling when it returns to its original height?

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44. Match the graphs (a-d) with the motion being graphed below. (Assume the object is initially stationary).

|a | |b | |

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| | |t | |

|stationary | | | |

|constant positive velocity | | | |

|constant positive acceleration | | | |

45. The graph of velocity versus time for a cart is given below.

vt (m/s)

|2 | | | | | | | |

| |t (s) |

a. Graph the acceleration of the cart as a function of time.

a (m/s2)

|1 | | | | | | | |

| |t (s) |

b. Graph the displacement of the cart as a function of time.

d (m)

|8 | | | | | | | |

| |t (s) |

Practice Multiple Choice

1. A car starting from rest accelerates uniformly at a rate of 5 m/s2. What is the car's speed after it has traveled 250 m?

(A) 20 m/s (B) 30 m/s (C) 40 m/s (D) 50 m/s

2. A ball is thrown straight downward with a speed of 0.5 m/s. What is the speed of the ball 0.70 s after it is released?

(A) 0.5 m/s (B) 10 m/s (C) 7.5 m/s (D) 15 m/s

3. A car increases its speed from 9.6 m/s to 11.2 m/s in 4 s. The average acceleration of the car during the 4 s is

(A) 0.4 m/s2 (B) 2.8 m/s2 (C) 2.4 m/s2 (D) 5.2 m/s2

4. What is the speed of an object after it has fallen freely from rest through a distance of 20 m?

(A) 5 m/s (B) 10 m/s (C) 20 m/s (D) 45 m/s

5. A car accelerates uniformly from rest, reaching a speed of 30 m/s in 6 s. During the 6 s, the car has traveled

(A) 15 m (B) 30 m (C) 60 m (D) 90 m

6. A student on her way to school walks four blocks east, three blocks north, and another four blocks east. Compared to the distance she walks, the magnitude of her displacement from home to school is

(A) less (B) greater (C) the same

7. An object is dropped from rest from the top of a high cliff. What is the distance the object falls during the first 6 s?

(A) 30 m (B) 60 m (C) 120 m (D) 180 m

8. A ball is dropped from the roof of a building 40 m tall. What is the approximate time of fall?

(A) 2.8 s (B) 4.1 s (C) 2.0 s (D) 8.2 s

9. A baseball is thrown upward with a speed of 30 m/s. The maximum height reached by the baseball is approximately

(A) 15 m (B) 75 m (C) 45 m (D) 90 m

10. A constant acceleration of 9.8 m/s2 on an object means the

(A) velocity increases 9.8 m/s during each second

(B) velocity is 9.8 m/s

(C) object falls 9.8 m during each second

(D) object falls 9.8 m during the first second only

11. An object is shot vertically upward. Which of the following correctly describes the velocity and acceleration of the object at its maximum elevation?

Velocity Acceleration

(A) Positive Positive

(B) Zero Zero

(C) Negative Negative

(D) Zero Negative

12. An object is released from rest on a planet that has no atmosphere. The object falls freely for 3 m in the first second. What is the planet's acceleration due to gravity?

(A) 1 m/s2 (B) 3 m/s2 (C) 6 m/s2 (D) 10 m/s2

13. Displacement x of an object as a function of time is shown.

[pic]

The acceleration of this object must be

14. The graph represents the relationship between speed and time for an object moving along a straight line.

[pic]

What is the distance traveled during the first 4 s?

(A) 5 m (B) 40 m (C) 20 m (D) 80 m

15. Which displacement/time graph best represents a cart traveling with a constant positive acceleration along a straight line?

(A) (B) (C) (D)

16. Which acceleration/time graph best represents an object falling freely near the earth's surface?

(A) (B) (C) (D)

17. Which of the following pairs of graphs shows the distance traveled versus time and the speed versus time for an object uniformly accelerated from rest at time t = 0?

(A) (B)

(C) (D)

18. A truck traveled 400 m north in 60 s, and then it traveled 300 m east in 40 s. The average velocity of the truck was

(A) 4 m/s (B) 5 m/s (C) 6 m/s (D) 7 m/s

19. The graph shows the velocity versus time for an object moving in a straight line.

[pic]

At what time after time = 0 does the object again pass through its initial position?

(A) 0.5 s (B) 1 s (C) 1.7 s (D) 2 s

Questions 20-21 At time t = 0, car X traveling with speed vo passes car Y, which is just starting to move. Both cars then travel on two parallel lanes of the same straight road. The graphs of speed v versus time t for both cars are shown.

|v (m/s) | | | | |

|2 vO | | | | |car Y |

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|vO | | | | |car X |

| | | | | | |

|0 | | | | |t (s) |

| |10 |30 | |

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(A) zero (B) constant but not zero

(C) increasing (D) decreasing

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