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Question #43476

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m = 2 \text{ kg}$ is moving along the positive x-axis. At $t = 0$ , its velocity is $v_0 = 2 \text{ m/s}$ . It experiences a net force in the direction of its motion given by $F = k v^2$ , where $k = 0.5 \text{ kg/m}$ and $v$ is its instantaneous velocity. What is the average power delivered by this force to the particle during the time interval from $t = 0$ to $t = 1 \text{ s}$ ?

Options

A.
$12 \text{ W}$
B.
$13.5 \text{ W}$
C.
$15 \text{ W}$
D.
$18 \text{ W}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Average vs. Instantaneous Calculus|Power-Velocity Relationship|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the distinction between instantaneous power ( $P = Fv$ ) and average power ( $P_{avg} = \Delta W / \Delta t$ ).

A common student trap is to assume that average power can be calculated by taking the arithmetic mean of the initial and final instantaneous powers ( $P_{avg} = (P_i + P_f)/2 = 18 \text{ W}$ ), or by evaluating the power function at the arithmetic mean of the velocities ( $P = k v_{mean}^3 = 13.5 \text{ W}$ ).

These algebraic shortcuts fail because the force and power are non-linear functions of velocity and time.

The rigorous approach requires integrating the equation of motion to find the final state, and then applying the Work-Energy Theorem to find the total work done over the interval before dividing by the time elapsed.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Average vs. Instantaneous Calculus
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free