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

Q1. Multiple Choice: Work, Energy, and Forces

Background

Topic: Basic concepts in work, energy, vectors, and forces.

This question tests your understanding of physical units, vector operations, 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 $|\vec{A}| |\vec{B}| \cos \theta$.

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

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

• Work by Gravity: $W = F_g \cdot d \cdot \cos \theta$; direction matters.

Step-by-Step Guidance

1. Review the definition of a Watt and recall which physical quantities it measures.

2. Recall the formula for the dot product between two vectors and match it to the options given.

3. Think about what it means for potential energy to be negative and whether this is possible in physics.

4. Consider the direction of the normal force relative to the displacement of the box to determine the work done by the normal force.

5. Analyze the direction of gravity relative to the elevator's motion to determine the sign of the work done by gravity.

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 analyze the motion of a basketball thrown vertically upward, including calculations of maximum height, gravitational potential energy, and 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: $KE_i + PE_i = KE_f + PE_f$

• Work by Gravity: $W_g = -mg\Delta h$ (if upward is positive)

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 initial kinetic and potential energies using the given values (mass, initial speed, initial height).

3. At maximum height, the ball's velocity is zero, so kinetic energy is zero; set up the equation to solve for the maximum height.

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

5. To find the work done by gravity, use $W_g = -mg\Delta h$, where $\Delta h$ is the change in height from the initial position to the maximum height.

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 involves a freight car compressing a spring, requiring you to calculate the potential energy stored in the spring, the car's initial kinetic energy, and its initial speed.

Key Terms and Formulas:

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

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

• Energy Conservation: $KE_{initial} = PE_{spring, max}$ (if 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 car's initial kinetic energy equal to the spring's potential energy at maximum compression.

3. Rearrange the kinetic energy formula to solve for the car's initial speed before hitting the spring.

Try solving on your own before revealing the answer!

Q4. Work and Power: Pulling a Wagon

Background

Topic: Work and Power

This problem asks you to calculate the work done by a force applied at an angle and the power output over a given time.

Key Terms and Formulas:

• Work: $W = Fd\cos\theta$

• Power: $P = \frac{W}{t}$

• Convert distance to meters if needed (1 cm = 0.01 m).

Step-by-Step Guidance

1. Convert the distance pulled from centimeters to meters.

2. Calculate the work done using the force, distance, and angle provided.

3. To find power, divide the work done 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 and Inelastic Collisions

This question involves analyzing a collision between two skaters, including drawing a before-and-after diagram, calculating initial momentum, and finding the final speed after collision.

Key Terms and Formulas:

• Momentum: $p = mv$

• Conservation of Momentum: $m_1v_1 + m_2v_2 = (m_1 + m_2)v_f$ (for perfectly inelastic collisions)

• Inelastic Collision: Objects stick together after collision.

Step-by-Step Guidance

1. Draw a diagram showing the positions and velocities of both skaters before and after the collision.

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

3. Set up the conservation of momentum equation to solve for the final speed of both skaters together.

Try solving on your own before revealing the answer!

Q6. Impulse and Momentum: Car Crash

Background

Topic: Impulse, Momentum, and Average Force

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

Key Terms and Formulas:

• Momentum: $p = mv$

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

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

Step-by-Step Guidance

1. Calculate the initial and final momentum of the driver (final velocity is zero after collision).

2. Find the impulse delivered by the seatbelt, which equals the change in momentum.

3. Calculate the average force using the impulse and the time interval of the collision.

Try solving on your own before revealing the answer!