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Chapter 7: Problem 42

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.

Short Answer

Expert verified
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 by step solution

01

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 \(KE_{initial}\).
02

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: \(m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}\) \(\frac{1}{2} m_1v_{1i}^2 + \frac{1}{2} m_2v_{2i}^2 = \frac{1}{2} m_1v_{1f}^2 + \frac{1}{2} m_2v_{2f}^2\)
03

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, \(v_{1f}\).

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Most popular questions from this chapter

In the movie Superman, Lois Lane falls from a building and is caught by the diving superhero. Assuming that Lois, with a mass of \(50.0 \mathrm{~kg}\), is falling at a terminal velocity of \(60.0 \mathrm{~m} / \mathrm{s}\), how much force does Superman exert on her if it takes \(0.100 \mathrm{~s}\) to slow her to a stop? If Lois can withstand a maximum acceleration of \(7 g^{\prime}\) s, what minimum time should it take Superman to stop her after he begins to slow her down?

Two balls of equal mass collide and stick together as shown in the figure. The initial velocity of ball \(\mathrm{B}\) is twice that of ball A. a) Calculate the angle above the horizontal of the motion of mass \(\mathrm{A}+\mathrm{B}\) after the collision. b) What is the ratio of the final velocity of the mass \(A+B\) to the initial velocity of ball \(A, v_{f} / v_{A} ?\) c) What is the ratio of the final energy of the system to the initial energy of the system, \(E_{\mathrm{f}} / E_{\mathrm{i}}\) ? Is the collision elastic or inelastic?

During an ice-skating extravaganza, Robin Hood on Ice, a 50.0 -kg archer is standing still on ice skates. Assume that the friction between the ice skates and the ice is negligible. The archer shoots a \(0.100-\mathrm{kg}\) arrow horizontally at a speed of \(95.0 \mathrm{~m} / \mathrm{s} .\) At what speed does the archer recoil?

A hockey puck with mass \(0.250 \mathrm{~kg}\) traveling along the blue line (a blue-colored straight line on the ice in a hockey rink) at \(1.50 \mathrm{~m} / \mathrm{s}\) strikes a stationary puck with the same mass. The first puck exits the collision in a direction that is \(30.0^{\circ}\) away from the blue line at a speed of \(0.750 \mathrm{~m} / \mathrm{s}\) (see the figure). What is the direction and magnitude of the velocity of the second puck after the collision? Is this an elastic collision?

A golf ball is released from rest from a height of \(0.811 \mathrm{~m}\) above the ground and has a collision with the ground, for which the coefficient of restitution is \(0.601 .\) What is the maximum height reached by this ball as it bounces back up after this collision?
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