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Question #42485

Difficulty: 8/10 | 🟢 Free

The Question

A block of mass $m$ is attached to a horizontal ideal spring of force constant $k$ and rests at equilibrium ( $x = 0$ ) on a frictionless surface. A bullet of mass $m$ moving with a horizontal velocity $v_0$ strikes the block and embeds within it instantaneously. The potential energy $U(x)$ of the spring-mass system is defined such that $U(x_0) = 0$ , where $x_0 \neq 0$ . If $E$ represents the total mechanical energy of the system after the collision, which of the following equations correctly describes the phase space trajectory ( $p$ vs $x$ ) of the resulting simple harmonic motion? (Here, $p$ is the momentum of the combined system).

Options

A.
$\frac{p^2}{4m\left(E + \frac{1}{2}kx_0^2\right)} + \frac{kx^2}{2\left(E + \frac{1}{2}kx_0^2\right)} = 1$
B.
$\frac{p^2}{2m\left(E + \frac{1}{2}kx_0^2\right)} + \frac{kx^2}{2\left(E + \frac{1}{2}kx_0^2\right)} = 1$
C.
$\frac{p^2}{4mE} + \frac{kx^2}{2E} = 1$
D.
$\frac{p^2}{4m\left(E - \frac{1}{2}kx_0^2\right)} + \frac{kx^2}{2\left(E - \frac{1}{2}kx_0^2\right)} = 1$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Potential Energy Reference|Phase Space Trajectory|Simple Pendulum Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental definition of phase space trajectories and potential energy reference frames.

Students often memorize the standard phase space ellipse equation $\frac{p^2}{2mE} + \frac{x^2}{2E/k} = 1$ , which implicitly assumes $U=0$ at the equilibrium position $x=0$ .

By shifting the zero-potential reference to $x=x_0$ , the total energy $E$ is offset by $-\frac{1}{2}kx_0^2$ .

Furthermore, the problem incorporates an inelastic collision to test if students correctly update the system's mass to $2m$ in the kinetic energy term $p^2/2M$ .

Memorizers will incorrectly select the standard formula or fail to account for the mass change.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Potential Energy Reference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free