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Chapter 9: Problem 73

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?

Short Answer

Expert verified
The ratio of their masses is 1:1 . They have equal masses.

Step by step solution

01

Setup the conservation of momentum equation

Let \( m \) be the mass of initially moving object and let \( M \) be the mass of the initially stationary object. Let the speed of the moving object before collision be \( u \), speed of \( m \) after collision be \( v \), and speed of \( M \) after collision be \( V \). According to conservation of momentum, we have \( m * u = m * v + M * V \).
02

Setup the conservation of kinetic energy equation

According to the problem statement, half of the kinetic energy of object \( m \) got transferred to object \( M \). So we have \(\frac{1}{2} * m * u^2 = \frac{1}{2} * m * v^2 + \frac{1}{2} * M * V^2\). After simplifying this, we get \( m * u^2 = m * v^2 + M * V^2 \).
03

Solving for mass ratio

Divide the equation from step 2 by the equation from step 1, we get \( u = v + \frac{M}{m} * V \). But from the equation in step 2 we know \( u = v + V \), therefore it implies that \( V = \frac{M}{m} * V \), which gives us the mass ratio \( M/m = 1 \). So the masses of the two objects are equal.

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