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Chapter 9: Problem 62

Two objects moving in opposite directions with the same speed \(v\) undergo a totally inelastic collision, and half the initial kinetic energy is lost. Find the ratio of their masses.

Short Answer

Expert verified
The ratio of the masses of the two objects \(\frac{m_1}{m_2}\) is 3.

Step by step solution

01

Establishing the Initial Energy

Firstly, analyze the initial energy. Both objects have the same speed \(v\) but in opposite directions. We should consider that the kinetic energy \(K\) for an object is given by \(\frac{1}{2} mv^2\). Here \(m\) is the mass of the object and \(v\) is its speed. As we have two objects with masses \(m_1\) and \(m_2\) both with speed \(v\), the total initial kinetic energy (\(K_{initial}\)) will be \(\frac{1}{2} m_1v^2 + \frac{1}{2} m_2v^2\).
02

Consider the Energy after Collision

After the collision, half of the initial kinetic energy is lost. Hence, the final kinetic energy (\(K_{final}\)) is only half of \(K_{initial}\), so \(K_{final} = \frac{1}{2} K_{initial} = \frac{1}{2} (\frac{1}{2} m_1v^2 + \frac{1}{2} m_2v^2)\).
03

Setup the Collision Result

After the collision, the two bodies move as one with a combined mass of \(m_1 + m_2\). As it's a totally inelastic collision, the kinetic energy is given by \(K_{final} = \frac{1}{2} (m_1 + m_2)V^2\), where \(V\) is the final speed.
04

Equate the Energies and Solve for the Ratio

Equate the equation found in Step 2 and Step 3, and solve for \(\frac{m_1}{m_2}\). \(\frac{1}{2} (m_1 + m_2)V^2 = \frac{1}{2} (\frac{1}{2} m_1v^2 + \frac{1}{2} m_2v^2)\) After simplification, the ratio \(\frac{m_1}{m_2}\) is 3.

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Most popular questions from this chapter

A block of mass \(m_{1}\) undergoes a one-dimensional elastic collision with an initially stationary block of mass \(m_{2} .\) Find an expression for the fraction of the initial kinetic energy transferred to the second block, and plot your result for mass ratios \(m_{1} / m_{2}\) from 0 to 20 .

You're working in quality control for a model rocket manufacturer, testing a class-D rocket whose specifications call for an impulse between 10 and \(20 \mathrm{N}\) \cdots. The rocket's burn time is \(\Delta t=2.8 \mathrm{s},\) and its thrust during that time is \(F(t)=a t(t-\Delta t)\) where \(a=-4.6 \mathrm{N} / \mathrm{s}^{2} .\) Does the rocket meet its specs?

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Two objects, one initially at rest, undergo a one-dimensional elastic collision. If half the kinetic energy of the initially moving object is transferred to the other object, what is the ratio of their masses?
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