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

Q1. Multiple Choice: Work, Energy, and Forces

Background

Topic: Fundamental concepts in mechanics, including units, vector operations, potential energy, and work done by forces.

This question tests your understanding of basic definitions and properties in physics, such as the units for work and energy, the dot product of vectors, the concept of 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.

• Work by Normal Force: The normal force acts perpendicular to the surface; work is $W = Fd\cos\theta$.

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

Step-by-Step Guidance

1. Review the definition of a Watt and recall which physical quantities it is used to measure.

2. Recall the formula for the dot product of two vectors and identify which option matches $|A||B|\cos\theta$.

3. Think about potential energy: Is it possible for it to be negative? Consider the choice of reference point.

4. For the box being pushed, analyze the direction of the normal force relative to the displacement to determine the work done.

5. For the elevator, consider 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 principle of conservation of mechanical energy to analyze the motion of a basketball thrown vertically upward. You'll need to consider kinetic and potential energy at different points in the motion.

Key Terms and Formulas:

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

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

• Conservation of Mechanical Energy: $KE_i + PE_i = KE_f + PE_f$

Step-by-Step Guidance

1. Identify the initial and final states: when the ball leaves the hand (initial) and at maximum height (final).

2. Write the conservation of energy equation, setting the ground as $PE = 0$.

3. At maximum height, the ball's velocity is zero, so $KE_f = 0$.

4. Set up the equation: $KE_i + PE_i = PE_f$ and solve for the maximum height.

5. For gravitational potential energy at maximum height, use $PE = mgh_{max}$.

6. To find the work done by gravity, use $W_{gravity} = -\Delta PE$ or consider the change in potential energy.

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. You'll use the concepts of kinetic energy, spring potential energy, and conservation of energy.

Key Terms and Formulas:

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

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

• Conservation of Energy: $KE_{initial} = PE_{spring, max}$ (if no friction)

Step-by-Step Guidance

1. Calculate the spring potential energy at maximum compression using $PE_{spring} = \frac{1}{2}kx^2$.

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

3. Use the kinetic energy formula to solve for the car's initial speed $v_0$.

4. Check units and ensure all quantities are in SI units (kg, m, s, N/m).

Try solving on your own before revealing the answer!

Q4. Work and Power: Pulling a Wagon

Background

Topic: Work and Power in Mechanics

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 and time to seconds as needed.

Step-by-Step Guidance

1. Convert the distance from centimeters to meters.

2. Calculate the work done using the formula $W = Fd\cos\theta$.

3. Convert the time from minutes to seconds for the power calculation.

4. Calculate the power output using $P = \frac{W}{t}$.

Try solving on your own before revealing the answer!

Q5. Ice Skaters: Conservation of Momentum

Background

Topic: Conservation of Linear Momentum in Collisions

This question involves a collision between two skaters, where one is initially moving and the other is stationary. After the collision, they move together, indicating a perfectly inelastic collision.

Key Terms and Formulas:

• Momentum: $p = mv$

• Conservation of Momentum: $m_1v_1 + m_2v_2 = (m_1 + m_2)v_f$

• Perfectly Inelastic Collision: Objects stick together after collision.

Step-by-Step Guidance

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

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 after collision.

4. Remember that the stationary skater's initial velocity is zero.

Try solving on your own before revealing the answer!

Q6. Car Crash: Impulse and Average Force

Background

Topic: Impulse, Momentum, and Average Force

This problem involves calculating the change in momentum (impulse) and the average force exerted during a collision.

Key Terms and Formulas:

• Change in Momentum: $\Delta p = m(v_f - v_i)$

• Impulse: $J = \Delta p$

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

Step-by-Step Guidance

1. Calculate the initial and final velocities of the driver (final velocity is zero after stopping).

2. Find the change in momentum using $\Delta p = m(v_f - v_i)$.

3. Impulse delivered by the seatbelt is equal to the change in momentum.

4. Calculate the average force using $F_{avg} = \frac{\Delta p}{\Delta t}$, where $\Delta t$ is the stopping time.

Try solving on your own before revealing the answer!