Problem 42 - University Physics with Modern Physics

Question

Two bumper cars moving on a frictionless surface collide elastically. The first bumper car is moving to the right with a speed of \(20.4 \mathrm{~m} / \mathrm{s}\) and rear-ends the second bumper car, which is also moving to the right but with a speed of \(9.00 \mathrm{~m} / \mathrm{s} .\) What is the speed of the first bumper car after the collision? The mass of the first bumper car is \(188 \mathrm{~kg}\), and the mass of the second bumper car is \(143 \mathrm{~kg}\). Assume that the collision takes place in one dimension.

Step-by-Step Solution

Answer

Short Answer: To find the final velocity of the first bumper car after an elastic collision, we first calculate the initial total momentum and kinetic energy of the system. Then, we apply the conservation of momentum and kinetic energy laws, resulting in two equations with two unknowns, \(v_{1f}\) and \(v_{2f}\). We solve these equations to get the final velocity of the first bumper car, \(v_{1f}\).

Step 1: Calculate initial momentum and kinetic energy

To start, we need to find the initial total momentum and kinetic energy of the system. The initial momentum (\(p_{initial}\)) and kinetic energy (\(KE_{initial}\)) can be calculated as follows: Initial momentum: \(p_{initial} = m_1v_{1i} + m_2v_{2i} = (188 \mathrm{~kg})(20.4 \mathrm{~m/s}) + (143 \mathrm{~kg})(9.00 \mathrm{~m/s})\) Initial kinetic energy: \(KE_{initial} = \frac{1}{2} m_1v_{1i}^2 + \frac{1}{2} m_2v_{2i}^2 = \frac{1}{2}(188 \mathrm{~kg})(20.4 \mathrm{~m/s})^2 + \frac{1}{2}(143 \mathrm{~kg})(9.00 \mathrm{~m/s})^2\) Calculate the values of \(p_{initial}\) and \(KE_{initial}\).

Step 2: Apply conservation of momentum and kinetic energy

Since the collision is elastic, both momentum and kinetic energy are conserved. We write the conservation equations for momentum (\(p_{final}\)) and kinetic energy (\(KE_{final}\)) as follows: Conservation of momentum: \(p_{final} = m_1v_{1f} + m_2v_{2f}\) Conservation of kinetic energy: \(KE_{final} = \frac{1}{2} m_1v_{1f}^2 + \frac{1}{2} m_2v_{2f}^2\) Using the fact that \(p_{initial} = p_{final}\) and \(KE_{initial} = KE_{final}\), we have the following equations: \(m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}\) \(\frac{1}{2} m_1v_{1i}^2 + \frac{1}{2} m_2v_{2i}^2 = \frac{1}{2} m_1v_{1f}^2 + \frac{1}{2} m_2v_{2f}^2\)

Step 3: Solve the equations for the final velocity of the first bumper car

Next, we need to solve the conservation equations to find \(v_{1f}\). We have two equations and two unknowns, \(v_{1f}\) and \(v_{2f}\). We can first solve for \(v_{2f}\) from the conservation of momentum equation: \(v_{2f} = \frac{m_1(v_{1i} - v_{1f}) + m_2v_{2i}}{m_2}\) Now, substitute this expression for \(v_{2f}\) into the conservation of kinetic energy equation and solve for \(v_{1f}\). After solving the equation, we will obtain the value for the final velocity of the first bumper car, \(v_{1f}\).

Key Concepts

Momentum Conservation

In physics, momentum conservation is a fundamental principle that applies in isolated systems, meaning no external forces influence the system. During any collision, such as the collision between two bumper cars, the total momentum of the system before the collision is equal to the total momentum after the collision.

Momentum is calculated using the formula:

• For an object, momentum (\(p\)) is the product of its mass (\(m\)) and velocity (\(v\)): \\(p = m \cdot v\)

In our example:

• The initial momentum of the system is the sum of the momenta of both bumper cars before they collide: \\(p_{initial} = m_1v_{1i} + m_2v_{2i}\)

After the collision, the principle of momentum conservation tells us:

• The total momentum remains the same: \\(m_1v_{1f} + m_2v_{2f} = p_{initial}\)

This conservation helps us find unknown velocities for the objects involved in the collision if their initial conditions and masses are known. It shows us how motion is transmitted between the colliding bodies in a predictable way.

Kinetic Energy Conservation

In elastic collisions, not only is momentum conserved, but kinetic energy is conserved as well. Kinetic energy is the energy that an object possesses due to its motion. The principle of kinetic energy conservation is especially simple yet powerful in describing these collisions.

The formula for kinetic energy (\(KE\)) of an object is:

• \(KE = \frac{1}{2} m v^2\)

For elastic collisions, the kinetic energy before and after the collision remains constant:

• Initial kinetic energy is the sum of the kinetic energies of both cars before collision: \\(KE_{initial} = \frac{1}{2} m_1v_{1i}^2 + \frac{1}{2} m_2v_{2i}^2\)
• After the collision, the total kinetic energy remains: \\(KE_{final} = \frac{1}{2} m_1v_{1f}^2 + \frac{1}{2} m_2v_{2f}^2\)

Kinetic energy conservation allows us to predict the post-collision speeds. This is crucial in determining how energies are redistributed during a perfectly elastic collision, where no energy is converted into other forms like sound or heat.

One-Dimensional Collisions

A one-dimensional collision is a collision where all movement happens along a single straight line. This type of collision simplifies analysis because we don't need to account for directions on more than one axis.

In the context of one-dimensional collisions, one can focus on:

• The velocity values are all scalar quantities, not requiring vector analysis
• All physical quantities like momentum and kinetic energy can be calculated with straightforward arithmetic using their linear components

For instance, the bumper cars in the problem collide along one line. This means we only consider their velocities in that single dimension, which are known both before and after the collision to apply respective conservation laws easily.

This approach often applies in simple physics problems to illustrate the basic principles of collisions, making them an excellent starting point for understanding more complex multidimensional cases.