Physics I: Work, Energy, Momentum, and Collisions – Guided Study Notes

Q1. Multiple Choice: Work, Energy, and Vectors

Background

Topic: Basic concepts in work, energy, and vector operations.

This question tests your understanding of physical units, vector dot products, potential energy, and the work done by forces in different scenarios.

Key Terms and Formulas:

• Watt (W): The SI unit of power, not work or energy. Work and energy are measured in Joules (J).

• Dot Product: For vectors $\vec{A}$ and $\vec{B}$, the dot product is $\vec{A} \cdot \vec{B} = |A||B|\cos\theta$.

• Potential Energy: Can be negative depending on the reference point chosen.

• Normal Force: Acts perpendicular to the surface; work done is $W = Fd\cos\theta$.

• Work by Gravity: $W = Fd\cos\theta$, where $\theta$ is the angle between force and displacement.

Step-by-Step Guidance

1. For the unit of work and energy, recall the SI units for each physical quantity (work, energy, power).

2. For the dot product, use the formula $\vec{A} \cdot \vec{B} = |A||B|\cos\theta$ and match it to the options given.

3. For potential energy, consider whether the reference point can be chosen such that potential energy is negative.

4. For the work done by the normal force, think about the direction of the normal force relative to the displacement of the box.

5. For the work done by gravity on an elevator moving upward, consider the direction of gravity relative to the elevator's motion.

Try solving on your own before revealing the answer!

Q2. Basketball Thrown Upward – Conservation of Energy

Background

Topic: Conservation of Mechanical Energy

This problem involves using the conservation of mechanical energy to determine the maximum height reached by a basketball, the gravitational potential energy at that height, and the work done by gravity.

Key Terms and Formulas:

• Kinetic Energy (KE): $KE = \frac{1}{2}mv^2$

• Gravitational Potential Energy (PE): $PE = mgh$

• Conservation of Mechanical Energy: $KE_{initial} + PE_{initial} = KE_{final} + PE_{final}$

• Work by Gravity: $W_{gravity} = -mg\Delta h$

Step-by-Step Guidance

1. Write the conservation of energy equation for the ball from the moment it leaves the player's hand to its maximum height.

2. Set the kinetic energy at the maximum height to zero (since the ball stops rising at that point).

3. Express the initial and final energies in terms of the given values (mass, initial speed, initial height, and unknown maximum height).

4. Rearrange the equation to solve for the maximum height (but do not calculate the final value yet).

5. For gravitational potential energy at maximum height, use $PE = mgh$ with the height found in the previous step.

Try solving on your own before revealing the answer!

Q3. Freight Car and Spring – Energy Conservation

Background

Topic: Conservation of Energy with Springs

This question asks you to analyze the energy transformations when a moving freight car compresses a spring and comes to rest.

Key Terms and Formulas:

• Spring Potential Energy: $PE_{spring} = \frac{1}{2}kx^2$

• Kinetic Energy: $KE = \frac{1}{2}mv^2$

• Energy Conservation: $KE_{initial} = PE_{spring, final}$ (assuming no friction)

Step-by-Step Guidance

1. Calculate the potential energy stored in the spring using the compression distance and spring constant.

2. Set the initial kinetic energy of the car equal to the spring's potential energy at maximum compression.

3. Rearrange the kinetic energy formula to solve for the car's initial speed (but do not compute the final value yet).

Try solving on your own before revealing the answer!

Q4. Work and Power: Pulling a Wagon

Background

Topic: Work and Power in Physics

This problem involves calculating the work done by a force applied at an angle and determining the power output over a given time.

Key Terms and Formulas:

• Work: $W = Fd\cos\theta$

• Power: $P = \frac{W}{t}$

Step-by-Step Guidance

1. Convert the distance from centimeters to meters for consistency in SI units.

2. Calculate the horizontal component of the force using $F\cos\theta$.

3. Multiply the horizontal force by the distance to find the work done.

4. For power, divide the work by the time taken (convert time to seconds if needed).

Try solving on your own before revealing the answer!

Q5. Collisions and Conservation of Momentum: Ice Skaters

Background

Topic: Conservation of Momentum in Collisions

This question explores the concepts of momentum, types of collisions, and how to analyze before-and-after scenarios.

Key Terms and Formulas:

• Momentum: $p = mv$

• Conservation of Momentum: $m_1v_1 + m_2v_2 = (m_1 + m_2)v_{final}$ (for perfectly inelastic collisions)

• Inelastic Collision: Objects stick together after collision.

Step-by-Step Guidance

1. Draw a before-and-after diagram showing the skaters' positions and velocities.

2. Identify the type of collision (elastic, inelastic, perfectly inelastic).

3. Calculate the initial momentum of the moving skater using $p = mv$.

4. Set up the conservation of momentum equation for the system before and after the collision.

5. Rearrange to solve for the final velocity of both skaters together (but do not compute the final value yet).

Try solving on your own before revealing the answer!

Q6. Impulse and Momentum: Car Crash

Background

Topic: Impulse, Momentum, and Forces in Collisions

This problem involves calculating the change in momentum, impulse, and average force during a car collision.

Key Terms and Formulas:

• Momentum: $p = mv$

• Impulse: $J = \Delta p = F_{avg} \Delta t$

• Change in Momentum: $\Delta p = m(v_{final} - v_{initial})$

• Average Force: $F_{avg} = \frac{\Delta p}{\Delta t}$

Step-by-Step Guidance

1. Calculate the initial and final momentum of the driver (remember the driver comes to rest).

2. Find the change in momentum using $\Delta p = m(v_{final} - v_{initial})$.

3. Impulse delivered by the seatbelt equals the change in momentum.

4. To find the average force, divide the impulse by the time interval of the collision.

Try solving on your own before revealing the answer!