TS EAMCET · Physics · Work Power Energy

A ball of mass \(2 \mathrm{~kg}\) is moving in \(x y\)-plane with a potential energy given as \(U=(12 x+16 y) \mathrm{J}, x\) and \(y\) being in metre. Assume the initial position of the ball at \(t=0\) is at origin \((0,0)\) and it is moving with a velocity of \((15 \hat{\mathbf{i}}+20 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}\). Then, identify the correct statement.

A The path of the ball is parabolic.
B The direction of motion of the ball initially at \(t=0\) is parallel to the direction of acceleration.
C The speed of the ball at \(t=2 \mathrm{~s}\) is \(5 \mathrm{~m} / \mathrm{s}\).
D The magnitude of acceleration of the ball is \(8 \mathrm{~m} / \mathrm{s}^2\).

Verified Solution

Answer & Solution

Correct Answer

(C) The speed of the ball at \(t=2 \mathrm{~s}\) is \(5 \mathrm{~m} / \mathrm{s}\).

Step-by-step Solution

Detailed explanation

Mass of the ball, \(m=2 \mathrm{~kg}\) Patential energy, \(U=(12 x+16 y) \mathrm{J}\) \(\therefore\) Force applied on it, \(\mathbf{F}=-\frac{d U}{d x} \hat{\mathbf{i}}-\frac{d U}{d y} \hat{\mathbf{j}}\) \[ \Rightarrow \quad \mathbf{F}=-12 \hat{\mathbf{i}}-16 \hat{\mathbf{j}} \]…

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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