Question:
An object of mass \(20\ \text{kg}\) is displaced by
\[
x=5t^2\ \text{m}
\]
(where \(t\) is time) by the application of a force. Then the ratio of the work done in times \(3\ \text{s}\) and \(5\ \text{s}\) is
An object of mass \(20\ \text{kg}\) is displaced by
\[
x=5t^2\ \text{m}
\]
(where \(t\) is time) by the application of a force. Then the ratio of the work done in times \(3\ \text{s}\) and \(5\ \text{s}\) is
Show Hint
If displacement is given as a function of time, first find velocity using
\[
v=\frac{dx}{dt}.
\]
Then use
\[
W=\Delta K=\frac{1}{2}mv^2
\]
to determine the work done.
Updated On: Jun 26, 2026
- \(\frac{2}{3}\)
- \(\frac{4}{9}\)
- \(\frac{3}{5}\)
- \(\frac{9}{25}\)
Show Solution
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The Correct Option is D
Solution and Explanation
Step 1: Find velocity as a function of time.
Given, \[ x=5t^2. \] Differentiating with respect to time, \[ v=\frac{dx}{dt} = 10t. \]
Step 2: Find the kinetic energy at time \(t\).
The mass of the object is \[ m=20\ \text{kg}. \] Therefore, \[ K=\frac{1}{2}mv^2. \] Substituting \(m=20\) and \(v=10t\), \[ K=\frac{1}{2}(20)(10t)^2. \] \[ K=10(100t^2). \] \[ K=1000t^2. \]
Step 3: Use the work-energy theorem.
Work done by the force is equal to the change in kinetic energy.
Since the particle starts from rest, \[ W=K=1000t^2. \] Thus, \[ W_{3}=1000(3)^2 = 9000. \] and \[ W_{5}=1000(5)^2 = 25000. \]
Step 4: Find the required ratio.
Therefore, \[ \frac{W_3}{W_5} = \frac{9000}{25000}. \] \[ = \frac{9}{25}. \]
Step 5: Final conclusion.
Hence, the required ratio is \[ \boxed{\frac{9}{25}} \] Therefore, the correct option is \[ \boxed{(4)} \]
Given, \[ x=5t^2. \] Differentiating with respect to time, \[ v=\frac{dx}{dt} = 10t. \]
Step 2: Find the kinetic energy at time \(t\).
The mass of the object is \[ m=20\ \text{kg}. \] Therefore, \[ K=\frac{1}{2}mv^2. \] Substituting \(m=20\) and \(v=10t\), \[ K=\frac{1}{2}(20)(10t)^2. \] \[ K=10(100t^2). \] \[ K=1000t^2. \]
Step 3: Use the work-energy theorem.
Work done by the force is equal to the change in kinetic energy.
Since the particle starts from rest, \[ W=K=1000t^2. \] Thus, \[ W_{3}=1000(3)^2 = 9000. \] and \[ W_{5}=1000(5)^2 = 25000. \]
Step 4: Find the required ratio.
Therefore, \[ \frac{W_3}{W_5} = \frac{9000}{25000}. \] \[ = \frac{9}{25}. \]
Step 5: Final conclusion.
Hence, the required ratio is \[ \boxed{\frac{9}{25}} \] Therefore, the correct option is \[ \boxed{(4)} \]
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