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Step-by-Step Solution
Step 1: State the Known Conditions
A body of mass $M$ is moving at speed $V_0$ and collides elastically with another body of mass $m$ which is initially at rest. After the collision, these masses move at angles $\theta_1$ and $\theta_2$ (measured from the original direction of motion). We are told that $\theta_1 = \theta_2$ at the largest possible value of the mass ratio $M/m$. We want to find that maximum ratio.
Step 2: Apply Conservation of Momentum
For an elastic collision, both momentum and kinetic energy are conserved.
Momentum Conservation (in the horizontal $x$-direction):
$$M\,V_0 = M\,v_1 \cos(\theta_1) + m\,v_2 \cos(\theta_2).$$
Momentum Conservation (in the vertical $y$-direction):
Because the second mass starts at rest and the motion initially has no $y$-component,
$$0 = M\,v_1 \sin(\theta_1) + m\,v_2 \sin(\theta_2),$$
assuming we define one of the directions positive and the other negative appropriately.
Step 3: Apply Conservation of Kinetic Energy
For an elastic collision, total kinetic energy before and after collision remains the same:
$$
\frac{1}{2}M\,V_0^2 = \frac{1}{2}M\,v_1^2 + \frac{1}{2}m\,v_2^2.
$$
Step 4: Impose the Condition $\theta_1 = \theta_2$
We set $\theta_1 = \theta_2 = \theta$. This simplifies the momentum equations considerably. By solving the system formed by the two momentum equations and the kinetic energy equation under the constraint $\theta_1 = \theta_2$, one arrives at the condition for the ratio $M/m$ that allows the angles to be equal.
Step 5: Conclude the Largest Possible Ratio
After carrying out the algebra (conserving both momentum and energy), the result shows that the largest possible value of the ratio $M/m$ for which the angles $\theta_1$ and $\theta_2$ can be equal is:
$$
\frac{M}{m} = 3.
$$
Final Answer:
The largest possible value of $M/m$ for which $\theta_1 = \theta_2$ is 3.
