TS EAMCET · Physics · Center of Mass Momentum and Collision

Hammer of mass \(M\) strikes a nail of mass \(m\) with a velocity \(20 \mathrm{~m} / \mathrm{s}\) into a fixed wall. The nail penetrates into the wall to a depth of \(1 \mathrm{~cm}\). The average resistance of the wall to the penetration of the nail is

A \(\left(\frac{M^2}{M+m}\right) \times 10^3\)
B \(\frac{2 M^2}{M+m} \times 10^4\)
C \(\frac{M+m}{M^2} \times 10^2\)
D \(\frac{M^2}{M+m} \times 10^2\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{2 M^2}{M+m} \times 10^4\)

Step-by-step Solution

Detailed explanation

From conservation of energy \[ \begin{aligned} M \times v+m(0) & =(M+m) v^{\prime} \\ v^{\prime} & =\frac{M \times 20}{(M+m)} \end{aligned} \] Average resistance of the wall to the penetration of the nail is \[ F=(M+m) \frac{\left(0^2+v^2\right)}{2 \times 10^{-2}} \]…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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