Physics Q31 – JEE-MAIN 2025

Home Physics Work, Energy & PowerQuestion #657

A block of mass 1 kg, moving along x with speed

v_{i} = 10

m/s enters a rough region ranging from

x = 0.1

m to

x = 1.9

m. The retarding force acting on the block in this range is

F_{r} = - k x

N, with

k = 10

N/m. Then the final speed of the block as it crosses rough region is.

Clue 1 of 3: Work-Energy Theorem

The change in kinetic energy of the block is equal to the total work done by all forces acting on it.

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Ninja Strategy•Initial Speed and Energy Loss Intuition

First, eliminate options that do not show a decrease in speed due to a retarding force. Then, use intuition about the magnitude of energy loss to rule out options that imply an unrealistically large or small change in kinetic energy.

🧠 Full Solution Path

Step 1: Calculate Work Done by Retarding Force✦ Active

The work done by the variable retarding force $F_{r} = - k x$ from $x_{1} = 0.1$ m to $x_{2} = 1.9$ m is given by the integral of force with respect to displacement. Given $k = 10$ N/m.

W = \int_{x_{1}}^{x_{2}} F_{r} d x = \int_{0.1}^{1.9} (- 10 x) d x

W = - 10 {[\frac{x^{2}}{2}]}_{0.1}^{1.9} = - 5 [(1.9)^{2} - (0.1)^{2}]

W = - 5 [3.61 - 0.01] = - 5 \times 3.60 = - 18 J

Step 2: Calculate Initial Kinetic Energy○ Expand

The initial kinetic energy of the block with mass $m = 1$ kg and initial speed $v_{i} = 10$ m/s is:

K E_{i} = \frac{1}{2} m v_{i}^{2} = \frac{1}{2} (1 kg) (10 m/s)^{2}

K E_{i} = \frac{1}{2} (1) (100) = 50 J

Step 3: Apply Work-Energy Theorem to find Final Speed○ Expand

According to the Work-Energy Theorem, the work done on the block equals the change in its kinetic energy ( $W = K E_{f} - K E_{i}$ ). We can use this to find the final kinetic energy and then the final speed $v_{f}$ .

W = K E_{f} - K E_{i}

- 18 J = \frac{1}{2} m v_{f}^{2} - 50 J

\frac{1}{2} (1) v_{f}^{2} = 50 - 18 = 32 J

v_{f}^{2} = 64 m^{2} / s^{2}

v_{f} = \sqrt{64} = 8 m/s

💡 Teacher's Secret Hint

Remember that speed is a scalar quantity and is always positive.

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Physics Question 31 – JEE-MAIN 2025

Clue 1 of 3: Work-Energy Theorem

🧠 Full Solution Path