Practice Questions: Kinematics

SECTION I: MULTIPLE CHOICE

Directions

Select the best answer for each question. You may use a calculator and reference sheet. Each question has exactly four answer choices. Answer all questions.

Questions 1-20

Question 1

Figure 1 shows the motion of an object moving along a straight line. Which of the following best describes the velocity of the object?

1. The velocity is zero because the object returns to its starting position.
2. The velocity is constant at 3 m/s.
3. The velocity increases uniformly from 0 m/s to 12 m/s.
4. The velocity is constant at 12 m/s.

Question 2

What is the magnitude of the car's acceleration?

1. 0.25 m/s²
2. 4.0 m/s²
3. 15 m/s²
4. 100 m/s²

Question 3

Based on Figure 2, during which time interval is the object decelerating?

1. 0 s to 3 s
2. 3 s to 7 s
3. The object never decelerates
4. 0 s to 7 s

Question 4

An object is thrown vertically upward with an initial velocity of 15 m/s from ground level. Ignoring air resistance and using \(g = 10\) m/s², approximately how long does it take for the object to reach its maximum height?

1. 0.67 s
2. 1.5 s
3. 3.0 s
4. 7.5 s

Question 5

Data Table:

The data table shows position versus time for a moving object. Which statement best characterizes the motion?

1. The object moves with constant velocity.
2. The object moves with increasing acceleration.
3. The object moves with constant acceleration.
4. The object moves with decreasing acceleration.

Question 6

A ball is dropped from rest from a height of 45 m above the ground. Using \(g = 10\) m/s² and ignoring air resistance, what is the speed of the ball just before it hits the ground?

1. 15 m/s
2. 30 m/s
3. 45 m/s
4. 90 m/s

Question 7

What is the cyclist's average velocity for the entire 40-second trip?

1. 0 m/s
2. 2.5 m/s north
3. 5.0 m/s north
4. 5.0 m/s south

Question 8

Based on Figure 3, what is the velocity of the object at \(t = 5\) seconds?

1. 2 m/s
2. 5 m/s
3. 10 m/s
4. 25 m/s

Question 9

A projectile is launched horizontally from a cliff with an initial velocity of 20 m/s. Ignoring air resistance, what is the horizontal velocity of the projectile after 3 seconds?

1. 0 m/s
2. 20 m/s
3. 30 m/s
4. 60 m/s

Question 10

At what time(s) is the instantaneous velocity of the object equal to zero?

1. At \(t = 0\) s and \(t = 4\) s only
2. At \(t = 2\) s only
3. At \(t = 0\) s, \(t = 2\) s, and \(t = 4\) s
4. The velocity is never zero

Question 11

An object moves along a straight line. Its position as a function of time is given by \(x(t) = 4t^2\), where \(x\) is in meters and \(t\) is in seconds. What is the average velocity of the object between \(t = 1\) s and \(t = 3\) s?

1. 4 m/s
2. 8 m/s
3. 16 m/s
4. 32 m/s

Question 12

Assuming constant acceleration, how far does the car travel during braking?

1. 31.25 m
2. 62.5 m
3. 125 m
4. 250 m

Question 13

Which statement correctly compares the displacement of the two objects from \(t = 0\) to \(t = 5\) s?

1. Object A travels twice as far as Object B.
2. Object B travels twice as far as Object A.
3. Both objects travel the same distance.
4. Object A travels three times as far as Object B.

Question 14

A stone is thrown straight downward from a bridge with an initial speed of 8 m/s. Using \(g = 10\) m/s² and ignoring air resistance, what is the stone's velocity after 2.0 seconds?

1. 12 m/s downward
2. 20 m/s downward
3. 28 m/s downward
4. 36 m/s downward

Question 15

During which time interval is the magnitude of the velocity greatest?

1. 0 s to 2 s
2. 2 s to 4 s
3. 4 s to 6 s
4. The magnitude is the same during 0-2 s and 4-6 s

Question 16

A projectile is launched at an angle of 30° above the horizontal with an initial speed of 40 m/s. Ignoring air resistance, what is the vertical component of the projectile's initial velocity?

1. 20 m/s
2. 28 m/s
3. 35 m/s
4. 40 m/s

Question 17

Data Set:

A student measures the time it takes for a ball to fall from various heights. The data are shown below:

Which graph would best linearize this data to determine the acceleration due to gravity?

1. Height vs. time
2. Height vs. time²
3. Height vs. 1/time
4. Height² vs. time

Question 18

At what time will Car B catch up to Car A?

1. 5 s
2. 10 s
3. 15 s
4. 20 s

Question 19

What is the displacement of the object from \(t = 0\) to \(t = 5\) s?

1. 20 m
2. 35 m
3. 50 m
4. 70 m

Question 20

An object is moving with constant acceleration along a straight line. At time \(t_1\), its velocity is \(v_1 = 5\) m/s. At time \(t_2 = t_1 + 4\) s, its velocity is \(v_2 = 17\) m/s. What is the displacement of the object during this time interval?

1. 22 m
2. 44 m
3. 68 m
4. 88 m

SECTION II: FREE RESPONSE

Directions

Answer both questions. Show all your work clearly for full credit. The suggested time for answering each question is provided. Begin each derivation by writing a fundamental physics principle or equation. Include appropriate units throughout your work.

FRQ 1: Mathematical Routines (Suggested time: 15 minutes)

1. Begin your derivation by writing a fundamental physics principle or equation. Derive an expression for the velocity \(v_1\) of the rocket at the end of the powered phase in terms of \(a\) and \(t_1\).
2. Derive an expression for the height \(h_1\) the rocket reaches at the end of the powered phase in terms of \(a\) and \(t_1\).
3. After the engines shut off, the rocket continues upward. Derive an expression for the additional height \(h_2\) the rocket gains after the engines shut off in terms of \(a\), \(t_1\), and \(g\).
4. Derive an expression for the total maximum height \(h_{total}\) reached by the rocket in terms of \(a\), \(t_1\), and \(g\).
5. Calculate the numerical value of \(h_{total}\) if \(a = 20\) m/s², \(t_1 = 5.0\) s, and \(g = 10\) m/s². Include appropriate units.

FRQ 2: Experimental Design and Analysis (Suggested time: 15 minutes)

1. Describe an experimental procedure the student could use to collect data relating drop height to time of fall. Your description should include:
   • What measurements should be taken
   • What equipment is needed
   • How to ensure accurate measurements
2. The relationship between drop height \(h\) and time of fall \(t\) for an object dropped from rest is given by \(h = \frac{1}{2}gt^2\), where \(g\) is the acceleration due to gravity. Describe how the student should graph the data to produce a linear relationship, and identify what the slope of the best-fit line represents.
3. The student collects the following data: Complete the third column of the data table by calculating \(t^2\) for each trial.
4. On the grid provided (or describe what you would draw), plot \(h\) versus \(t^2\) and draw a best-fit line through the data points.
5. Using your best-fit line, calculate the experimental value of the acceleration due to gravity \(g\). Show all work and include units.

ANSWER KEY

Part A: Multiple Choice Answer Table

Part B: FRQ Detailed Answers

FRQ 1 - Answer Key

Part A: Derive velocity at end of powered phase

Begin with fundamental principle:

\[ v = v_0 + at \]

Since the rocket starts from rest, \(v_0 = 0\).
During the powered phase, the acceleration is \(a\) and the time interval is \(t_1\).
Substituting into the equation:

\[ v_1 = 0 + at_1 \]

\[ \boxed{v_1 = at_1} \]

Part B: Derive height at end of powered phase

Use the kinematic equation:

\[ h = v_0 t + \frac{1}{2}at^2 \]

Since \(v_0 = 0\), the acceleration is \(a\), and the time is \(t_1\):

\[ h_1 = 0 + \frac{1}{2}a t_1^2 \]

\[ \boxed{h_1 = \frac{1}{2}at_1^2} \]

Part C: Derive additional height after engines shut off

At the start of the unpowered phase:
Initial velocity: \(v_1 = at_1\)
Acceleration: \(-g\) (downward)
Final velocity at maximum height: \(v_f = 0\)

Use the kinematic equation:

\[ v_f^2 = v_1^2 + 2(-g)h_2 \]

Substituting \(v_f = 0\) and \(v_1 = at_1\):

\[ 0 = (at_1)^2 - 2gh_2 \]

\[ 2gh_2 = a^2t_1^2 \]

\[ h_2 = \frac{a^2t_1^2}{2g} \]

\[ \boxed{h_2 = \frac{a^2t_1^2}{2g}} \]

Part D: Derive total maximum height

The total maximum height is the sum of the two heights:

\[ h_{total} = h_1 + h_2 \]

Substituting the expressions from Parts B and C:

\[ h_{total} = \frac{1}{2}at_1^2 + \frac{a^2t_1^2}{2g} \]

Factor out common terms:

\[ h_{total} = \frac{at_1^2}{2}\left(1 + \frac{a}{g}\right) \]

Or equivalently:

\[ \boxed{h_{total} = \frac{at_1^2}{2}\left(\frac{g + a}{g}\right)} \]

Or in expanded form:

\[ \boxed{h_{total} = \frac{at_1^2(g + a)}{2g}} \]

Part E: Calculate numerical value of total maximum height

Given:
\(a = 20\) m/s²
\(t_1 = 5.0\) s
\(g = 10\) m/s²

Substitute into the expression for \(h_{total}\):

\[ h_{total} = \frac{(20\text{ m/s}^2)(5.0\text{ s})^2(10\text{ m/s}^2 + 20\text{ m/s}^2)}{2(10\text{ m/s}^2)} \]

\[ h_{total} = \frac{(20)(25)(30)}{20} \text{ m} \]

\[ h_{total} = \frac{15000}{20} \text{ m} \]

\[ h_{total} = 750 \text{ m} \]

\[ \boxed{h_{total} = 750 \text{ m}} \]

FRQ 2 - Answer Key

Part A: Describe experimental procedure

A complete procedure should include:

• Measurements to be taken: Measure the drop height \(h\) from the bottom of the ball to the ground using a meterstick. Use a motion detector or timer to measure the time of fall \(t\) from release until the ball hits the ground.
• Equipment needed: Meterstick or measuring tape, ball, motion detector or photogate timer, stands or clamps to hold measuring equipment, level surface.
• Procedure steps: Set up the meterstick vertically. Mark several heights (e.g., 1 m, 2 m, 3 m, 4 m, 5 m). Position the motion detector directly below the drop point. Drop the ball from rest from each marked height. Record the time of fall for each height. Repeat each trial at least three times and calculate the average time for each height to reduce measurement uncertainty.
• Ensuring accuracy: Make sure the ball is released from rest (zero initial velocity). Ensure the motion detector is properly calibrated. Drop the ball straight down without giving it any horizontal motion. Use a level to ensure the meterstick is vertical.

Part B: Describe how to graph data and identify slope

Graphing method:

The equation \(h = \frac{1}{2}gt^2\) is of the form \(y = mx\) when written as \(h = \left(\frac{g}{2}\right)t^2\).

To linearize the data, the student should plot drop height \(h\) on the vertical axis and \(t^2\) on the horizontal axis.

What the slope represents:

The slope of the best-fit line is equal to \(\frac{g}{2}\), where \(g\) is the acceleration due to gravity. Therefore, \(g = 2 \times \text{slope}\).

Part C: Complete the data table

Calculations:
\(0.50^2 = 0.25\)
\(0.70^2 = 0.49\)
\(1.00^2 = 1.00\)
\(1.20^2 = 1.44\)
\(1.42^2 = 2.02\)

Part D: Plot and draw best-fit line

Description of graph:

On a coordinate grid with \(t^2\) (s²) on the horizontal axis and \(h\) (m) on the vertical axis, plot the five data points: (0.25, 1.25), (0.49, 2.45), (1.00, 5.00), (1.44, 7.20), (2.02, 10.00).

Draw a straight line that passes through or near the origin and best fits the trend of the data points. The line should have nearly equal numbers of points above and below it.

Part E: Calculate experimental value of \(g\)

Determine the slope of the best-fit line:

Using two points on the best-fit line (ideally endpoints for accuracy), for example (0, 0) and (2.00, 10.00):

\[ \text{slope} = \frac{\Delta h}{\Delta t^2} = \frac{10.00\text{ m} - 0\text{ m}}{2.00\text{ s}^2 - 0\text{ s}^2} \]

\[ \text{slope} = \frac{10.00}{2.00} = 5.00 \text{ m/s}^2 \]

Calculate \(g\):

Since slope \(= \frac{g}{2}\):

\[ g = 2 \times \text{slope} \]

\[ g = 2 \times 5.00\text{ m/s}^2 \]

\[ \boxed{g = 10.0 \text{ m/s}^2} \]

Note: The experimental value of \(g\) is 10.0 m/s², which is consistent with the commonly used approximation for acceleration due to gravity.