Physics 1: Kinematics—One Dimension Name __________________________A. Measurement (1-4 to 1-6)

1. science knowledge is advanced by observing patterns (laws) and constructing explanations (theories), which are supported by repeatable experimental evidencea. theory lasts until disprovenb. theory is never 100 % certain

2. uncertainty in measurementsa. precision and accuracy

1. precise = consistent (even if incorrect)2. accurate = correct (even if inconsistent)

b. data analysis1. accuracy is measured by percent difference

% = 100|mean – true|/true 2. precision is measured by percent deviation

% = 100|trial – mean|/N(mean)(N is number of trials)

c. significant figures (sf) indicate level of certainty

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 significant2. zero is sometimes significant, sometimes not

a. example: 0.00053000021000 never always ?

b. (?) decimal vs. no decimal1. 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 calculations1. limited by least accurate measurement2. x, : answer has the same number of sf as

the measurement with the fewest3. +, –: answer has same end decimal position

as measurement with left most end position3. SI measuring system

a. summary chartMeasurement 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 mb. m km:

455 m x 1 km/103 m = 0.455 km2. 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 technique455 kg x 10 3 g x (10 -2 ) 3 m 3 = 0.455 g m3 1 kg cm3 cm3

4. scientific notation: C x 10n

a. conversion from decimal to scientific notation1. 1,200,000 = 1.2 x 106

2. 0.0000012 = 1.2 x 10-6

b. significant figures 1. C contains only significant figures2. 1200 with 3 significant figures = 1.20 x 103

B. Data Analysis Using Graphs (2-8)1. graphing data (i.e. position "x" vs. time "t")

a. Cartesian axis1. x-axis is independent variable (t)2. y-axis is the dependent variable (x)

b. axis labels1. 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 subscripts1. subscript o (xo) indicate starting position

(usually the origin: xo = 0)2. subscript t (xt) indicate position at time “t”

(usually the t is dropped: xt = x)d. + vs. –: direction can be ahead (+) or behind (–)

2. graphing 2-dimentional positiona. 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, ) 1. r is distance from the origin2. is angle measured from +x 3. translation to Cartesian coordinates

a. x = rcosb. y = rsin

3. best fit line (i.e. position "x" vs. time "t")a. individual coordinates may not line up exactly

because of small experimental errorsb. 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 errors3. data that is well outside best fit line should be

repeatedc. interpreting graphs

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

d. values can be determined from the graph1. speed (velocity) = position/time: (v = x/t)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 (x)

4. graph sequenceslope area

slope area

y = x y' = k y" = 0

y = x2 y' = x y" = k

5. slope of a curve = slope of tangent line

tangent line

C. Kinematics (2-1 to 2-7)1. displacement (distance): x = d = x – xo (m)2. change in time: t = t – to (s) (usually to = 0 t = 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 = (v – vo)/t (m/s2)a. instantaneous velocity, vt, is velocity at time, t

special case: if vo = 0, then v = 2vav

b. falling objects1. 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 casea. direction can be "forward (+) or backward (–)b. velocity and displacement always have the same

signc. 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 problemsa. constant motion (a = 0)

draw diagram complete chart with two numbers and one letter

d vav t

use the definition of average velocity: vav = d/tb. accelerated motion (a 0)

draw diagram complete chart with given information (three numbers

and two letters)d vo v a t

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

Unused Letter Formulaa d = ½(vo + v)td v = vo + atv d = vot + ½at2

t v2 = vo2 + 2ad

8. deriving the kinematic formulas a. d = area under a v vs. t graph

d = x – xo = vavt = ½(vo + v)t = vot + ½at2

v v

vav at

vo

vo t

b. rearrange definition for a = (v – vo)/t v = vo + at c. algebraic rearrangement of formula 1 and 2

Steps Algebrasolve for t in formula 1solve for t in formula 2set equal to each othercross multiplefoilrearrange

t = 2d/(vo + v)t = (v – vo)/a2d/(vo + v) = (v – vo)/a2ad = (v + vo)(v – vo)2ad = v2 – vo

2

v2 = v02 + 2ad

Experiments1. Best Timing Method

Time how long it takes the swinging pendulum bob to swing through the lowest point and return using the two techniques and determine the best person and technique.a. Collect the following data on your performance.Sight: Watch the bob to start and stop the timer.

TimeTrial 1 Trial 2 Trial 3 Mean

Sound: Listen for "start" and "stop" (without watching) to start and stop the timer.

TimeTrial 1 Trial 2 Trial 3 Mean

b. Calculate the following from the data.Formula

(true time = 1.00 s)Calculation

Sight Sound

% % = 100|mean – 1.00| 1.00

% % = 100 |trial – mean| N(mean)

c. Which is yourmost accurate method (smallest % )?most precise method (smallest % )?d. Who in your group is the overall best timer (smallest %

+ % ). What method did he/she use?Best personBest method

2. Graphing Slope and Area LabDrop the cylinder from the distances listed and measure the final velocity using the velocimeter, graph the data and determine slope and area.a. Collect the following data.

Distance (m) 0.20 0.31 0.44 0.60 0.78 1.00 1.22

Vel

ocity

(km

/hr) Trial 1

Trial 2

Trial 3

Mean

Velocity (m/s)Time (s) 0.20 0.25 0.30 0.35 0.40 0.45 0.50b. Use the grid below to graph velocity (m/s) vs. time (s).

(1) Label the axis with appropriate scales and units(2) Indicate the data points with a small circle.(3) Draw a best fit straight line using a ruler. (use the

origin as a data point)

c. Determine the slope of the best fit line.

d. The formula that relates velocity, v, with acceleration due to gravity, g, is, v = gt, where g is the slope of the line. What is the value of g? (Include units)

e. The actual (true) acceleration due to gravity is 9.8 m/s2. What is the percent difference?

f. What is the area between the line from 0 s to 0.50 s and the x-axis?

g. The formula for that relates distance, d, with time, t, is d = vt, where d is the area under the line. How far did the cylinder drop in 0.50 s?

h. The expected velocities at each time are listed below. Determine the average percent difference between the expected and lab velocities.

Velocity (m/s)Time (s) 0.20 0.25 0.30 0.35 0.40 0.45 0.50Expected v 2.0 2.5 2.9 3.4 3.9 4.4 4.9Measured v%

Average %

3. Graphing Constant Velocity LabAdjust the track for constant 1.44 km/hr (0.400 m/s) velocity, measure the time it takes for the bearing to travel the distances below, graph the data, and calculate the percent difference between slope and known velocity.a. Collect the following data.

d (m) 0 0.20 0.40 0.60 0.80

Time(s)

Average 0

b. Graph the data and draw a best-fit line. d (m)

0.8

0.6

0.4

0.2

0.0 0.4 0.8 1.2 1.6 2.0 t (s)

c. Determine the velocity from the slope of the line.

d. Calculate the percent difference between the true velocity of 0.400 m/s and the velocity from part c.

4. Graphing Acceleration LabConfigure the track for acceleration, measure the time it takes for the bearing to travel the distances below, graph the data, and determine the slope and area. a. Calculate the following from the data.

d (m) 0 0.20 0.40 0.60 0.80

Time(s)

Average 0b. Calculate the following from the data.

distances 0.20 0.40 0.60 0.80

vav vav = d/tav

v v = 2vav

a a = v/tav

c. Graph d vs. t. Draw a best-fit, curved line (include 0,0). d (m)

0.8

0.6

0.4

0.2

0 1 2 3 4t (s)

d. Graph v vs. t. Draw a best-fit, straight line (include 0,0). v (m/s)

0.5

0.4

0.3

0.2

0.1

0 1 2 3 4t (s)

e. Graph a vs. t. Draw a best-fit, horizontal line. a (m/s2)

0.4

0.3

0.2

0.1

0 1 2 3 4t (s)

f. Determine acceleration from the graphs.

Slope of v vs. t graph (include units)

y-intercept of a vs. t graph

% Deviation

g. Determine displacement from the graphs.Area from 0 to 3 s under v vs. t graph (include units)

y-value at 3 s on d vs. t graph

% Deviation

h. Design a procedure for determining acceleration using the velocimeter. Collect the data below.d (m) 0.20 0.40 0.60 0.80

v (km/hr)j. Calculate the following from the data.

v v = vkm/hr/3.6

a v2 = 2ad

aaverage

Practice ProblemsA. Measurement

1. Express the height of a 5'4" person using correct sf in inches

(1' = 12")centimeters

(1" = 2.54 cm)meters

(1 m = 100 cm)

2. A student makes repeated masses of an object. a. Complete the chart to determine the % deviation.

Mass 48.307 g 49.886 g 50.911 g 49.524 g

Mean

Deviation

Average Deviation

% b. The actual (true) mass of the object is 50.000 g. What

is the percent error?

c. Were these measurements more precise than accurate or more accurate than precise? Explain

B. Data Analysis Using Graphs3. A student measured the weight Fs needed to stretch a

spring by the following lengths x.x (m) 0 .05 0.10 0.15 0.20 0.25 0.30 0.40Fs (N) 0 2.50 7.0 12.0 16.5 19.0 25.0 32.0a. Use the grid to graph the data.

(1) Indicate the data points with small circles.(2) Draw a best fit straight line using a ruler.

Fs (N)

30

20

10

0 0.1 0.2 0.3 0.4x (m)

b. Determine the slope of the best fit line.

c. The formula that governs spring behavior is Fs = kx. What is the value of k for this spring? (Include units)

d. What is the area under the line from 0 m to 0.40 m?

e. The formula for work is, W = Fx. How much work is needed to stretch this spring 0.40 m? (include units)

4. A pilot flies a distance of 110 km at a heading of 25o North of East. Determine the x and y coordinates of the destination.

C. Kinematics5. 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?

b. What is the displacement?

c. What is the person's average velocity?

d. What is the person's average speed?

19. A person runs from 50 m to 30 m in 2 s.a. What is the runner's displacement?

b. What is the runner's velocity?

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

b. Determine the runner's average speed.

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

22. Determine the acceleration for the following situations.a. A car initially at rest is traveling 15 m/s 5 s later.

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

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 v a t

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

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

d vo v a t

Determine the minimum length of runway.

25. A rock is dropped from a 100 m cliff. d vo v a t

Determine the time that the rock is in the air.

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

d vo v a t

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

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

How high is the cliff?

28. A car accelerates from 10 m/s to 30 m/s in 10 s.d vo v a t

a. What is the car's acceleration?

b. How far did the car travel during acceleration?

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

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

The breaks can decelerate the car at -6.0 m/s2. d vo v a t

b. How far does the car travel during deceleration?

c. What is the total distance traveled?

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.

Bill accelerates from rest at 2.5 m/s2.d vo v a t

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

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

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

b. Complete the list of variables for the highest point. d vo v a t

(1) How high does the ball rise?

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

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 MotionQuestions 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 C35. 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)

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

b. Calculate the acceleration for each time interval.8 s to 10 s 18 s to 20 s

c. Graph the velocity as a function of time. v (m/s)

2

1

0

-1

-2

-3 4 8 12 16 20t (s)

39. The graph of velocity versus time for a cart is given below. v (m/s)

4

2

0

-2

2 4 6 8 10t (s)

a. At what times was the cart at rest?

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

c. Determine the acceleration for each time interval.0 to 2 s 2 s to 3 s

3 s to 5 s 5 s to 7 s

7 s to 11 s 11 s to 12 s

d. Graph the acceleration of the cart as a function of time. a (m/s2)

1

0

-1

2 4 6 8 10t (s)

e. Graph the displacement of the cart as a function of time. d (m)

6

4

2

0

-2

-4

-6

2 4 6 8 10t (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?

b. How far does he travel during the acceleration?

At the next stop light he decelerates to a stop at 4 m/s2. c. How much time does this take?

d. How far does he travel during deceleration?

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

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?

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?

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?

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

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

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

b. How high does the ball rise?

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

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

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

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

a b

t

c

t

d

t t

motion graphed d vs. t v vs. t a vs. tstationaryconstant positive velocityconstant positive acceleration

45. The graph of velocity versus time for a cart is given below. vt (m/s)

2

0

-2

2 4 6 8 10t (s)

a. Graph the acceleration of the cart as a function of time. a (m/s2)

1

0

-1

2 4 6 8 10t (s)

b. Graph the displacement of the cart as a function of time. d (m)

8

6

4

2

0

-22 4 6 8 10

t (s)

Practice Multiple ChoiceBriefly explain why the answer is correct in the space provided.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.

The acceleration of this object must be (A) zero (B) constant but not zero(C) increasing (D) decreasing

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

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.

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

vO car X

0 t (s)10 3020. Which is true at t = 20 s?

(A) Car X is ahead (B) Car X is passing car Y (C) Car Y is ahead (D) Car Y is passing car X

21. At what time is car Y just passing car X? (A) 0 s (B) 20 s (C) 30 s (D) 40 s

22. An object released from rest at time t = 0 slides down a frictionless incline a distance of 1 m during the first second.

The distance traveled by the object during the time interval from t = 1 s to t = 2 s is (A) 1 m (B) 2 m (C) 3 m (D) 4 m

23. The graph shows velocity v versus time t for an object.

Which is the graph of position x versus time t? (A) (B)

(C) (D)

24. A body moving in the positive x direction passes the origin at time t = 0. Between t = 0 and t = 1 s, the body has a constant speed of 24 m/s. At t = 1 s, the body is given a constant acceleration of -6 m/s2 (in the negative x direction). The position x of the body at t = 11 s is (A) +99 m (B) +36 m (C) –36 m (D) –75 m

Questions 25-26 The graph represents position x versus time t for an object accelerating from rest with constant acceleration.

25. The average speed during the interval between 0 s and 2 s is most nearly (A) 2 m/s (B) 4 m/s (C)6 m/s (D) 8 m/s

26. The instantaneous speed at 2 s is most nearly(A) 2 m/s (B) 4 m/s (C)6 m/s (D) 8 m/s

Practice Free Response1. The graph of velocity versus time for a cart is given below.

vt (m/s)

4

2

0

-2

0 2 4 6 8 10t (s)

Determine the time or time interval for the following.Cart is moving away from originCart is stationaryCart is moving toward the originCart has returned to originWhen farthest from the originPositive accelerationZero accelerationNegative acceleration

2. A 95-m-long train begins uniform acceleration from rest. The front of the train has a speed of 25 m/s when it passes a railway worker who is standing 180 m from where the front of the train started.

What will be the speed of the last car as it passes the worker?

3. A model rocket is launched vertically with an acceleration of 30 m/s2 for 2 s. The rocket continues upward until it reaches its highest point, when a parachute is deployed. The rocket descends vertically to the ground at 5 m/s.a. How high is the rocket after the acceleration stage?

b. How fast is the rocket traveling at the end of the acceleration stage?

c. What is the rocket's maximum height?

d. At what time will the maximum height be reached?

e. How long does it take the rocket to descend?

4. A toy cart, initially at rest, starts to move down an incline with constant acceleration. When the cart reaches an arbitrary point, its position x is measured for different times t and the data are recorded in the table.a. Calculate the average speed of the cart during each

0.50 s time interval and fill in the spaces in the table. Time, t (s) 0.00 0.50 1.00 1.50 2.00Position, x (m) 0.00 0.22 0.70 1.35 2.10Velocity, vav (m/s)b. Label the vertical axis with appropriate numbers, plot

the data, and draw a best-fit line. v (m/s)

1.4

1.0

0.8

0.4

0 0.4 0.8 1.2 1.6t (s)

c. Using the best-fit line to determine(1) the slope (acceleration).

(2) the y-intercept (initial velocity).

d. At the end of 2.00 s, determine(1) the velocity of the cart.

(2) the distance traveled from rest.

5. A person jumps from 15.0 m above a firefighter's safety net. The survivor stretches the net 1.0 m before coming to rest.

a. What was the average deceleration experienced by the survivor when she was slowed to rest by the net?

b. What would you do to make it safer (generate smaller acceleration), would you stiffen or loosen the net? Explain your reasoning.