Question

A particle at rest on a frictionless table is acted upon by a horizontal force which is constant in magnitude and direction. A graph is plotted for the work done on the particle \(W\) against the speed of the particle \(v\). If there are no frictional forces acting on the particle, the graph will look like:

• A. Option a
• B. Option b
• C. Option c
• D. Option d

Hint:

Key relations to remember:
Work–Energy theorem: \(W = \Delta K\)
\(K = \frac{1}{2}mv^2\)
Hence, \(W\) vs \(v\) graph is parabolic Always check whether the motion starts from rest.

Answer 1

The Correct Option is D

Step 1: Apply the work–energy theorem. The work done by the force on the particle is equal to the change in kinetic energy: \[ W = \Delta K \] Since the particle starts from rest, \[ W = \frac{1}{2}mv^2 \]
Step 2: Express work as a function of velocity. \[ W \propto v^2 \] This shows that work done varies as the square of the speed.
Step 3: Analyse the nature of the graph.
At \(v=0\), \(W=0\)
As \(v\) increases, \(W\) increases non-linearly
The curve is a parabola opening upward
Step 4: Match with the given options.
(a) Linear graph — incorrect
(b) Constant work — incorrect
(c) Saturating curve — incorrect
(d) Upward curving parabola — correct Final Answer: \[ \boxed{\text{Option (d)}} \]