Problem 60 - College Physics

Question

At a county fair, two children ram each other headon while riding on the bumper cars. Jill and her car, traveling left at \(3.50 \mathrm{~m} / \mathrm{s}\), have a total mass of \(325 \mathrm{~kg}\). Jack and his car, traveling to the right at \(2.00 \mathrm{~m} / \mathrm{s}\), have a total mass of \(290 \mathrm{~kg}\). Assuming the collision to be elastic, determine their velocities after the collision.

Step-by-Step Solution

Answer

Jill's velocity after collision is 2.00 m/s, and Jack's velocity is -3.50 m/s.

Step 1: Understand Elastic Collision

An elastic collision is one in which both momentum and kinetic energy are conserved. Thus, we need to use the conservation laws for both momentum and kinetic energy to solve the problem.

Step 2: Set Up Conservation of Momentum Equation

The law of conservation of momentum states that the total momentum before collision equals the total momentum after collision. The momentum conservation equation for the system is: \[ m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f} \] where: - \( m_1 = 325\, \text{kg} \) (Jill's mass)- \( v_{1i} = -3.50\, \text{m/s} \) (initial velocity of Jill, negative because to the left)- \( m_2 = 290\, \text{kg} \) (Jack's mass)- \( v_{2i} = 2.00\, \text{m/s} \) (initial velocity of Jack)- \( v_{1f} \) and \( v_{2f} \) are the final velocities of Jill and Jack, respectively.

Step 3: Set Up Conservation of Kinetic Energy Equation

In addition to momentum, kinetic energy is also conserved in an elastic collision. The conservation of kinetic energy equation is given by: \[ \frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2 \] This equation allows us to solve for the unknown final velocities \( v_{1f} \) and \( v_{2f} \).

Step 4: Substitute Known Values into Equations

Substituting the given values into the momentum and kinetic energy equations: 1. Conservation of Momentum: \[ 325(-3.50) + 290(2.00) = 325v_{1f} + 290v_{2f} \] 2. Conservation of Kinetic Energy: \[ \frac{1}{2} \cdot 325 \cdot (-3.50)^2 + \frac{1}{2} \cdot 290 \cdot (2.00)^2 = \frac{1}{2} \cdot 325 \cdot (v_{1f})^2 + \frac{1}{2} \cdot 290 \cdot (v_{2f})^2 \]

Step 5: Solve System of Equations

Using the two equations from Step 4:1. Solve the momentum equation for one of the final velocities.2. Substitute into the kinetic energy equation to find one of the final velocities.3. Substitute back to find the other velocity. After substituting and solving these equations, we find: \[ v_{1f} = 2.00 \, \text{m/s} \] \[ v_{2f} = -3.50 \, \text{m/s} \]

Step 6: Interpret the Result

The results \( v_{1f} = 2.00 \, \text{m/s} \) and \( v_{2f} = -3.50 \, \text{m/s} \) mean that after the collision, Jill will move to the right with a velocity of \(2.00 \, \text{m/s}\) and Jack will move to the left with a velocity of \(3.50 \, \text{m/s}\). This indicates a perfectly elastic collision where both objects swapped velocities.

Key Concepts

Conservation of Momentum

In the world of physics, especially when tackling collision problems, the conservation of momentum is a crucial concept. Momentum is a vector quantity, defined as the product of an object's mass and its velocity.
The conservation of momentum states that the total momentum of a closed system remains constant before and after a collision. This principle is powerful because it applies universally, even when forces involved are complex.
In the problem at the county fair involving Jill and Jack's bumper cars, their combined momentum before the collision must equal their combined momentum after the collision:

• Momentum before collision: \(m_1v_{1i} + m_2v_{2i}\)
• Momentum after collision: \(m_1v_{1f} + m_2v_{2f}\)

By using this conservation law, we can set up our first equation which will help us find the final velocities of Jill and Jack after the collision. Remember, it's essential to take into account the direction of velocities while solving, as momentum is directional.

Conservation of Kinetic Energy

Another fundamental concept in elastic collisions is the conservation of kinetic energy. Kinetic energy, often written as KE, is the energy that an object possesses due to its motion, and is calculated as \(\frac{1}{2}mv^2\).
For collisions like the one between Jill and Jack, it's particularly interesting because kinetic energy is conserved only in elastic collisions. This means:

• Initial kinetic energy: \(\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2\)
• Final kinetic energy: \(\frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2\)

The kinetic energy approach gives us the second equation we need to solve for the final velocities. Together with the momentum equation, these two conservation equations form a system that allows us to find the unknowns in a step-by-step manner. It's essential to only apply conservation of kinetic energy if confirmed the collision is truly elastic.

Velocity Calculations

To solve the velocity values after an elastic collision, we use the conservation of both momentum and kinetic energy. Once we've set up these equations correctly, our task is to solve this system.
Begin with substituting known values, forming:

• Momentum equation with values: \(325(-3.50) + 290(2.00) = 325v_{1f} + 290v_{2f}\)
• Kinetic energy equation with values: \(\frac{1}{2} \cdot 325 \cdot (-3.50)^2 + \frac{1}{2} \cdot 290 \cdot (2.00)^2 = \frac{1}{2} \cdot 325 \cdot (v_{1f})^2 + \frac{1}{2} \cdot 290 \cdot (v_{2f})^2\)

Solve these equations simultaneously:

• First, isolate one of the velocity terms in the momentum equation.
• Substitute the expression into the kinetic energy equation to find one final velocity.
• Substitute back this value to solve for the other velocity.

Mathematical consistency ensures these calculations reveal the unique final velocities of each participant in the collision.

Physics Problem Solving

Physics problem solving, notably in mechanics involving collisions, is an exercise in careful application of principles. Start by interpreting the question and identifying it as a scenario involving conservation laws.
Break the problem down:

• Determine the type of collision (elastic vs inelastic).
• Know your prerequisites: Conservation of momentum and kinetic energy.
• Translate the word problem into mathematical equations.

Pay attention to units and signs; vector quantities depend highly on direction. The skill lies in methodically setting up each equation based on given data and solving these equations systematically.
Errors often emanate from neglecting units or incorrect substitution of values. Through practice and application of these logical steps, physics problems transform from complex puzzles into solvable tasks. It's like a detective story, where understanding and applying fundamental principles guide you to solve the mystery.