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Question #42039

Difficulty: 8/10 | 🟢 Free

The Question

A truck of initial mass $M_0 = 1000 \text{ kg}$ is moving with velocity $v_0 = 45 \text{ m/s}$ on a horizontal road. The coefficient of kinetic friction between the truck and the road is $\mu = 0.5$ . The truck's cargo leaks vertically downwards at a constant rate $\lambda = 20 \text{ kg/s}$ . If the deceleration process is governed by the generalized Newton's law $F_{\text{ext}} = \frac{dp}{dt}$ (where $p$ is the instantaneous momentum of the truck and its remaining cargo), calculate the total distance traveled by the truck before coming to rest. (Take $g = 10 \text{ m/s}^2$ )

Options

A.
$1125 - 4000 \ln(1.25) \text{ m}$
B.
$202.5 \text{ m}$
C.
$1125 - 2000 \ln(1.25) \text{ m}$
D.
$2250 - 4000 \ln(1.25) \text{ m}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Stopping Distance Paradox|Variable Mass Systems|Differential Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the application of the generalized Newton's Second Law ( $F = dp/dt$ ) for a variable mass system.

The common trap is assuming $F = m(t)a$ , which leads to a constant deceleration $a = -\mu g$ and an incorrect stopping distance of $v_0^2 / 2\mu g = 202.5$ m.

By forcing the use of $F = dp/dt$ , students must account for the $v(dm/dt)$ term representing the momentum flux of the leaking cargo.

This creates a non-linear deceleration profile where the decreasing mass reduces the frictional force, competing with the changing momentum, ultimately requiring the solution of a first-order differential equation and integration to find the true stopping distance.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Stopping Distance Paradox
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free