Physics with Calculus: Acceleration and Free Fall Study Guidance

Q1. A car's velocity changes from 4 m/s to 24 m/s in 5 s. Find its acceleration.

Background

Topic: Acceleration (Kinematics)

This question tests your understanding of how to calculate acceleration when given the initial and final velocities and the time interval.

Key Terms and Formulas

• Acceleration ($a$): The rate of change of velocity with respect to time.

• Initial velocity ($v_i$): The starting velocity of the object.

• Final velocity ($v_f$): The velocity of the object at the end of the time interval.

• Time ($t$): The duration over which the velocity changes.

Key formula:

$a = \frac{v_f - v_i}{t}$

Step-by-Step Guidance

1. Identify the known values: $v_i = 4\ \mathrm{m/s}$, $v_f = 24\ \mathrm{m/s}$, $t = 5\ \mathrm{s}$.

2. Write the formula for acceleration: $a = \frac{v_f - v_i}{t}$.

3. Substitute the known values into the formula, but do not calculate yet.

4. Set up the subtraction in the numerator and the value for the denominator.

Try solving on your own before revealing the answer!

Final Answer: 4 m/s²

$a = \frac{24 - 4}{5} = \frac{20}{5} = 4\ \mathrm{m/s^2}$

The car's acceleration is 4 meters per second squared.

Q2. A ball is dropped from rest. What is its speed after 4 s?

Background

Topic: Free Fall (Kinematics)

This question tests your ability to use the equations of motion for objects in free fall, starting from rest.

Key Terms and Formulas

• Free-fall acceleration ($g$): The acceleration due to gravity, approximately $10\ \mathrm{m/s^2}$ downward.

• Initial velocity ($v_i$): For a dropped object, $v_i = 0$.

• Time ($t$): The duration the object has been falling.

Key formula:

$v = gt$

Step-by-Step Guidance

1. Identify the known values: $g = 10\ \mathrm{m/s^2}$, $t = 4\ \mathrm{s}$, $v_i = 0$.

2. Write the formula for speed after falling from rest: $v = gt$.

3. Substitute the values for $g$ and $t$ into the formula.

Try solving on your own before revealing the answer!

Final Answer: 40 m/s

$v = (10\ \mathrm{m/s^2})(4\ \mathrm{s}) = 40\ \mathrm{m/s}$

The ball's speed after 4 seconds is 40 meters per second downward.

Q3. How far does a dropped ball fall in 2 s?

Background

Topic: Free Fall Distance (Kinematics)

This question tests your ability to calculate the distance an object falls under constant acceleration due to gravity, starting from rest.

Key Terms and Formulas

• Distance ($d$): The vertical distance fallen.

• Initial velocity ($v_i$): For a dropped object, $v_i = 0$.

• Time ($t$): The time the object has been falling.

• Acceleration due to gravity ($g$): $10\ \mathrm{m/s^2}$.

Key formula:

$d = \frac{1}{2}gt^2$

Step-by-Step Guidance

1. Identify the known values: $g = 10\ \mathrm{m/s^2}$, $t = 2\ \mathrm{s}$, $v_i = 0$.

2. Write the formula for distance fallen from rest: $d = \frac{1}{2}gt^2$.

3. Substitute the values for $g$ and $t$ into the formula, but do not compute yet.

4. Set up the multiplication and exponentiation.

Try solving on your own before revealing the answer!

Final Answer: 20 m

$d = \frac{1}{2} \times 10 \times (2)^2 = 5 \times 4 = 20\ \mathrm{m}$

The ball falls 20 meters in 2 seconds.

Q4. A car slows from 25 m/s to 5 m/s in 4 s. Find its acceleration.

Background

Topic: Acceleration (Kinematics, Deceleration)

This question tests your ability to calculate acceleration (which will be negative here, indicating slowing down) given initial and final velocities and time.

Key Terms and Formulas

• Acceleration ($a$): The rate of change of velocity.

• Initial velocity ($v_i$): $25\ \mathrm{m/s}$.

• Final velocity ($v_f$): $5\ \mathrm{m/s}$.

• Time ($t$): $4\ \mathrm{s}$.

Key formula:

$a = \frac{v_f - v_i}{t}$

Step-by-Step Guidance

1. Identify the known values: $v_i = 25\ \mathrm{m/s}$, $v_f = 5\ \mathrm{m/s}$, $t = 4\ \mathrm{s}$.

2. Write the formula for acceleration: $a = \frac{v_f - v_i}{t}$.

3. Substitute the values into the formula, but do not calculate yet.

4. Set up the subtraction in the numerator and the value for the denominator.

Try solving on your own before revealing the answer!

Final Answer: -5 m/s²

$a = \frac{5 - 25}{4} = \frac{-20}{4} = -5\ \mathrm{m/s^2}$

The negative sign indicates the car is slowing down (decelerating).

Q5. A ball is thrown straight upward. Describe its velocity and acceleration at the highest point.

Background

Topic: Free Fall, Motion at the Top of Trajectory

This question tests your conceptual understanding of what happens to the velocity and acceleration of an object at the peak of its upward motion.

Key Terms and Formulas

• Velocity ($v$): The speed and direction of the object.

• Acceleration ($a$): The rate of change of velocity, due to gravity ($g$).

• At the highest point: The instant when the object changes direction from upward to downward.

Step-by-Step Guidance

1. Recall that at the highest point, the ball's velocity is momentarily zero because it stops rising before falling back down.

2. Remember that acceleration due to gravity ($g$) is always acting downward, even at the top.

3. State the values for velocity and acceleration at this instant, but do not write the final answer yet.

Try solving on your own before revealing the answer!

Final Answer:

At the highest point, the ball's velocity is $0\ \mathrm{m/s}$, but its acceleration is still $10\ \mathrm{m/s^2}$ downward (due to gravity).

This is a key concept in projectile motion: acceleration due to gravity is constant throughout the motion.

Q6. On a velocity-time graph, velocity increases from 10 m/s at 2 s to 30 m/s at 6 s. Find the acceleration.

Background

Topic: Acceleration from Velocity-Time Graph

This question tests your ability to calculate acceleration as the slope of a velocity-time graph.

Key Terms and Formulas

• Acceleration ($a$): Slope of the velocity-time graph.

• Change in velocity ($\Delta v$): $v_f - v_i$.

• Change in time ($\Delta t$): $t_f - t_i$.

Key formula:

$a = \frac{\Delta v}{\Delta t}$

Step-by-Step Guidance

1. Identify the initial and final velocities: $v_i = 10\ \mathrm{m/s}$ at $t_i = 2\ \mathrm{s}$, $v_f = 30\ \mathrm{m/s}$ at $t_f = 6\ \mathrm{s}$.

2. Calculate $\Delta v = v_f - v_i$ and $\Delta t = t_f - t_i$.

3. Write the formula for acceleration: $a = \frac{\Delta v}{\Delta t}$.

4. Substitute the values into the formula, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Final Answer: 5 m/s²

$a = \frac{30 - 10}{6 - 2} = \frac{20}{4} = 5\ \mathrm{m/s^2}$

The acceleration is 5 meters per second squared.

Q7. A rock is dropped from a bridge and hits the water 3 s later. Estimate the height of the bridge.

Background

Topic: Free Fall Distance (Kinematics)

This question tests your ability to calculate the vertical distance an object falls under gravity, starting from rest, given the time of fall.

Key Terms and Formulas

• Distance ($d$): The height of the bridge.

• Time ($t$): $3\ \mathrm{s}$.

• Acceleration due to gravity ($g$): $10\ \mathrm{m/s^2}$.

• Initial velocity ($v_i$): $0$ (dropped from rest).

Key formula:

$d = \frac{1}{2}gt^2$

Step-by-Step Guidance

1. Identify the known values: $g = 10\ \mathrm{m/s^2}$, $t = 3\ \mathrm{s}$, $v_i = 0$.

2. Write the formula for distance fallen from rest: $d = \frac{1}{2}gt^2$.

3. Substitute the values for $g$ and $t$ into the formula, but do not compute yet.

4. Set up the multiplication and exponentiation.

Try solving on your own before revealing the answer!

Final Answer: 45 m

$d = \frac{1}{2} \times 10 \times (3)^2 = 5 \times 9 = 45\ \mathrm{m}$

The bridge is approximately 45 meters high.

Q8. A ball is thrown upward at 20 m/s. Approximately how long does it take to reach the top?

Background

Topic: Free Fall, Time to Maximum Height

This question tests your ability to determine the time it takes for an object thrown upward to reach its highest point, using kinematic equations.

Key Terms and Formulas

• Initial velocity ($v_i$): $20\ \mathrm{m/s}$ upward.

• Final velocity at the top ($v_f$): $0\ \mathrm{m/s}$ (momentarily at rest).

• Acceleration due to gravity ($g$): $10\ \mathrm{m/s^2}$ downward.

• Time to reach the top ($t$): What you are solving for.

Key formula:

$v_f = v_i - gt$ (since upward is positive and gravity acts downward)

Step-by-Step Guidance

1. Set $v_f = 0$ at the top of the trajectory.

2. Write the formula: $0 = v_i - gt$.

3. Rearrange the equation to solve for $t$.

4. Substitute $v_i = 20\ \mathrm{m/s}$ and $g = 10\ \mathrm{m/s^2}$, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Final Answer: 2 s

$0 = 20 - 10t \implies t = \frac{20}{10} = 2\ \mathrm{s}$

It takes 2 seconds for the ball to reach the top of its trajectory.

Q9. Velocity-Time Graph: Calculate the slope using rise/run. Show work and include units.

Background

Topic: Acceleration from Velocity-Time Graph

This question tests your ability to determine acceleration as the slope of a velocity-time graph, using given data points.

Key Terms and Formulas

• Acceleration ($a$): Slope of the velocity-time graph.

• Rise: Change in velocity ($\Delta v$).

• Run: Change in time ($\Delta t$).

Key formula:

$a = \frac{\Delta v}{\Delta t}$

Step-by-Step Guidance

1. Choose two points from the table, for example: $(t_1, v_1) = (0, 0)$ and $(t_2, v_2) = (5, 50)$.

2. Calculate the rise: $\Delta v = v_2 - v_1$.

3. Calculate the run: $\Delta t = t_2 - t_1$.

4. Write the formula for the slope: $a = \frac{\Delta v}{\Delta t}$, and substitute the values, but do not compute yet.

Try solving on your own before revealing the answer!

Final Answer: 10 m/s²

$a = \frac{50 - 0}{5 - 0} = \frac{50}{5} = 10\ \mathrm{m/s^2}$

The slope of the velocity-time graph, which is the acceleration, is 10 meters per second squared.

Q10. Use the graph to estimate the velocity at 4.5 s.

Background

Topic: Interpreting Velocity-Time Graphs

This question tests your ability to estimate values from a linear velocity-time graph, using the relationship between time and velocity for constant acceleration.

Key Terms and Formulas

• Velocity ($v$): The value you are estimating at a specific time.

• Time ($t$): $4.5\ \mathrm{s}$.

• Acceleration ($a$): $10\ \mathrm{m/s^2}$ (from previous question).

• Initial velocity ($v_0$): $0\ \mathrm{m/s}$ (dropped from rest).

Key formula:

$v = v_0 + at$

Step-by-Step Guidance

1. Identify the initial velocity ($v_0 = 0$) and acceleration ($a = 10\ \mathrm{m/s^2}$).

2. Write the formula for velocity at time $t$: $v = v_0 + at$.

3. Substitute $t = 4.5\ \mathrm{s}$ into the formula, but do not compute yet.

Try solving on your own before revealing the answer!

Final Answer: 45 m/s

$v = 0 + (10)(4.5) = 45\ \mathrm{m/s}$

The estimated velocity at 4.5 seconds is 45 meters per second.