Physics I: Kinematics, Dynamics, and Motion Study Guide

Problem 1: Kinematic Plots

Q1a. What was Marty’s average acceleration after the entire 20 seconds?

Background

Topic: Kinematics – Average Acceleration

This question tests your understanding of how to calculate average acceleration from a velocity vs. time graph.

Key Terms and Formulas

• Average acceleration ($a_{avg}$): The change in velocity divided by the change in time.

$a_{avg} = \frac{v_f - v_i}{t_f - t_i}$

Step-by-Step Guidance

1. Identify the initial velocity ($v_i$) at $t = 0$ seconds from the graph.

2. Identify the final velocity ($v_f$) at $t = 20$ seconds from the graph.

3. Calculate the change in velocity: $\Delta v = v_f - v_i$.

4. Calculate the change in time: $\Delta t = t_f - t_i = 20 - 0$ seconds.

Try solving on your own before revealing the answer!

Q1b. What was Marty’s acceleration at $t = 2.5$ seconds?

Background

Topic: Instantaneous Acceleration

This question tests your ability to determine the instantaneous acceleration from a velocity vs. time graph, which is the slope at a specific point.

Key Terms and Formulas

• Instantaneous acceleration: The slope of the velocity vs. time graph at a specific time.

$a = \frac{dv}{dt}$

Step-by-Step Guidance

1. Locate $t = 2.5$ seconds on the time axis of the graph.

2. Estimate the slope of the velocity curve at this point (rise over run).

3. Use two nearby points to calculate $\frac{\Delta v}{\Delta t}$ around $t = 2.5$ s.

Try solving on your own before revealing the answer!

Q1c. What was Marty’s acceleration at $t = 10$ seconds?

Background

Topic: Instantaneous Acceleration

This is similar to part (b), but at a different time. You are again finding the slope of the velocity vs. time graph at $t = 10$ s.

Key Terms and Formulas

• Instantaneous acceleration: $a = \frac{dv}{dt}$

Step-by-Step Guidance

1. Find $t = 10$ seconds on the graph.

2. Determine the slope of the velocity curve at this point (use two points close to $t = 10$ s).

3. Calculate $\frac{\Delta v}{\Delta t}$ for those points.

Try solving on your own before revealing the answer!

Q1d. At what time does Marty stop and turn around?

Background

Topic: Kinematics – Turning Point

This question tests your ability to interpret a velocity vs. time graph to find when the object changes direction (velocity crosses zero).

Key Terms and Formulas

• Turning point: When velocity $v = 0$ and changes sign.

Step-by-Step Guidance

1. Look for the point on the graph where the velocity curve crosses the time axis ($v = 0$).

2. Read the corresponding time value from the graph.

Try solving on your own before revealing the answer!

Q1e. Sketch the position versus time graph for Marty’s motion.

Background

Topic: Position-Time Graphs from Velocity-Time Graphs

This question tests your ability to qualitatively sketch a position vs. time graph based on the shape of a velocity vs. time graph.

Key Concepts

• When velocity is positive, position increases; when velocity is negative, position decreases.

• The slope of the position vs. time graph at any point is the velocity at that time.

• Where velocity crosses zero, the position graph has a maximum or minimum (turning point).

Step-by-Step Guidance

1. Identify intervals where velocity is positive, negative, or zero.

2. Sketch a curve that increases when velocity is positive and decreases when velocity is negative.

3. Mark the turning point where velocity crosses zero.

4. Label your axes: position (y-axis), time (x-axis).

Try sketching the graph before checking the answer!

Problem 2: 1-D Motion

Q2a. If car #1 starts at the origin ($x = 0$), at what position do the two cars run into each other?

Background

Topic: Kinematics – Relative Motion with Constant Acceleration

This question tests your ability to set up and solve equations for two objects moving toward each other with constant acceleration.

Key Terms and Formulas

• Position with constant acceleration: $x = x_0 + v_0 t + \frac{1}{2} a t^2$

Step-by-Step Guidance

1. Write the position equation for car #1: $x_1 = 0 + 0 \cdot t + \frac{1}{2} a_1 t^2$

2. Write the position equation for car #2, starting at $x = 50$ m and moving toward car #1: $x_2 = 50 + 0 \cdot t - \frac{1}{2} a_2 t^2$

3. Set $x_1 = x_2$ to find the time when they meet.

4. Solve for $t$ using the given accelerations.

5. Plug $t$ back into either position equation to find the meeting position.

Try solving on your own before revealing the answer!

Q2b. If $a_3 = \alpha t$, what would the position equation for car #3 look like?

Background

Topic: Kinematics – Non-Constant Acceleration

This question tests your ability to integrate acceleration as a function of time to find velocity and position.

Key Terms and Formulas

• Acceleration: $a_3 = \alpha t$

• Velocity: $v = \int a \, dt$

• Position: $x = \int v \, dt$

Step-by-Step Guidance

1. Integrate $a_3 = \alpha t$ with respect to $t$ to find $v_3(t)$.

2. Integrate $v_3(t)$ with respect to $t$ to find $x_3(t)$.

3. Include constants of integration if initial velocity or position are not zero.

Try solving on your own before revealing the answer!

Problem 3: Circular Motion and Newton’s 2nd Law

Q3a. If mass #1 is experiencing 12 g’s of acceleration, how fast is it travelling?

Background

Topic: Circular Motion – Centripetal Acceleration

This question tests your understanding of centripetal acceleration and how to relate it to speed in circular motion.

Key Terms and Formulas

• Centripetal acceleration: $a_c = \frac{v^2}{r}$

• Given: $a_c = 12g$, $g = 9.81$ m/s$^2$, $r = 8$ m

Step-by-Step Guidance

1. Calculate the total centripetal acceleration: $a_c = 12 \times 9.81$ m/s$^2$.

2. Set $a_c = \frac{v^2}{r}$ and solve for $v$.

3. Plug in the values for $a_c$ and $r$ to find $v$.

Try solving on your own before revealing the answer!

Q3b. How fast is mass #2 travelling?

Background

Topic: Circular Motion – Linear Speed at Different Radii

This question tests your understanding that objects at different radii in circular motion have different linear speeds if they complete a revolution in the same time.

Key Terms and Formulas

• Linear speed: $v = \omega r$

• Angular velocity ($\omega$) is the same for both masses.

Step-by-Step Guidance

1. Find the angular velocity $\omega$ using $v_1$ and $r_1$ from part (a): $\omega = \frac{v_1}{r_1}$.

2. Calculate $v_2 = \omega r_2$, where $r_2 = 10$ m (8 m + 2 m rod).

Try solving on your own before revealing the answer!

Q3c. How many times does mass 2 travel a full circle in one minute?

Background

Topic: Circular Motion – Frequency and Period

This question tests your ability to relate linear speed, circumference, and time to find the number of revolutions per minute.

Key Terms and Formulas

• Circumference: $C = 2\pi r$

• Number of revolutions: $N = \frac{v \cdot t}{C}$

Step-by-Step Guidance

1. Calculate the circumference for mass 2: $C = 2\pi r_2$.

2. Use the speed $v_2$ from part (b) and $t = 60$ s to find the number of revolutions: $N = \frac{v_2 \cdot 60}{C}$.

Try solving on your own before revealing the answer!

Problem 4: Relative Velocity

Q4a. Draw a diagram depicting this problem.

Background

Topic: Relative Velocity in Two Dimensions

This question tests your ability to visualize and represent velocity vectors for objects moving in different directions.

Key Concepts

• Draw vectors for the boat, river, and car, labeling their directions and magnitudes.

Step-by-Step Guidance

1. Draw an x-y coordinate system.

2. Draw the boat's velocity vector: $10 \hat{x} + 5 \hat{y}$ m/s.

3. Draw the river's velocity vector: $-3 \hat{x}$ m/s.

4. Draw the car's velocity vector: $-30 \hat{x}$ m/s.

Try sketching the diagram before checking the answer!

Q4b. What would the velocity of the boat be if the river were calm?

Background

Topic: Relative Velocity – Reference Frames

This question tests your understanding of how to subtract the river's velocity to find the boat's velocity relative to the water.

Key Terms and Formulas

• Relative velocity: $\vec{v}_{B,rel} = \vec{v}_{B,obs} - \vec{v}_{R,obs}$

Step-by-Step Guidance

1. Write the observed velocity of the boat: $\vec{v}_B = 10 \hat{x} + 5 \hat{y}$ m/s.

2. Write the river's velocity: $\vec{v}_R = -3 \hat{x}$ m/s.

3. Subtract the river's velocity from the boat's observed velocity to get the boat's velocity in a calm river.

Try solving on your own before revealing the answer!

Q4c. What would the magnitude of the boat’s velocity in a calm river and what is the angle that would be measured from the x-axis?

Background

Topic: Vector Magnitude and Direction

This question tests your ability to calculate the magnitude and direction (angle) of a vector from its components.

Key Terms and Formulas

• Magnitude: $|\vec{v}| = \sqrt{v_x^2 + v_y^2}$

• Angle: $\theta = \arctan\left(\frac{v_y}{v_x}\right)$

Step-by-Step Guidance

1. Use the x and y components of the boat's velocity in a calm river from part (b).

2. Calculate the magnitude using the Pythagorean theorem.

3. Calculate the angle from the x-axis using the arctangent formula.

Try solving on your own before revealing the answer!

Problem 5: Projectile Motion

Q5a. Draw a diagram depicting the situation.

Background

Topic: Projectile Motion

This question tests your ability to visualize projectile motion, including the launch angle and range.

Key Concepts

• Draw the cannon at the origin, the projectile's parabolic path, and label the launch angle ($60^\circ$) and range (60 m).

Step-by-Step Guidance

1. Draw an x-y coordinate system with the cannon at the origin.

2. Draw the projectile's path as a parabola, landing at $x = 60$ m.

3. Label the launch angle $\theta = 60^\circ$ from the x-axis.

Try sketching the diagram before checking the answer!

Q5b. What speed did the projectile leave the cannon at?

Background

Topic: Projectile Motion – Range Equation

This question tests your ability to use the range equation for projectile motion to solve for the initial speed.

Key Terms and Formulas

• Range equation: $R = \frac{v_0^2 \sin 2\theta}{g}$

• Given: $R = 60$ m, $\theta = 60^\circ$, $g = 9.81$ m/s$^2$

Step-by-Step Guidance

1. Write the range equation for projectile motion: $R = \frac{v_0^2 \sin 2\theta}{g}$.

2. Plug in the known values for $R$, $\theta$, and $g$.

3. Rearrange the equation to solve for $v_0$.

4. Calculate $\sin 2\theta$ for $\theta = 60^\circ$.

Try solving on your own before revealing the answer!