A body of mass 4 kg is placed on a plane at a point P having coordinate $(3,4) \mathrm{m}$. Under the action of force $\overrightarrow{\mathrm{F}}=(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}) \mathrm{N}$, it moves to a new point Q having coordinates $(6,10) \mathrm{m}$ in 4 sec . The average power and instantaneous power at the end of 4 sec are in the ratio of :

JEE Main · 2025 · 28 Jan 2025 · Shift 2 · Work, Power & Energy

A body of mass 4 kg is placed on a plane at a point P having coordinate

(3, 4) m

. Under the action of force

F = (2 \hat{i} + 3 \hat{j}) N

, it moves to a new point Q having coordinates

(6, 10) m

in 4 sec . The average power and instantaneous power at the end of 4 sec are in the ratio of :

Answer: (B)

Average power:

The displacement from P(3,4) to Q(6,10) is

s = (6 - 3) \hat{i} + (10 - 4) \hat{j} = 3 \hat{i} + 6 \hat{j} m

Work done by the force

F = (2 \hat{i} + 3 \hat{j}) N

W = F \cdot s = 2 \times 3 + 3 \times 6 = 6 + 18 = 24 J

Average power over the 4 s interval:

⟨ P ⟩ = \frac{W}{t} = \frac{24}{4} = 6 W

Instantaneous power at t = 4 s:

Acceleration of the 4 kg body:

a = \frac{F}{m} = \frac{2 i ^ + 3 j ^}{4} = \frac{1}{2} \hat{i} + \frac{3}{4} \hat{j} m/s^{2}

Starting from rest, the velocity at

t = 4 s

v = a t = (\frac{1}{2} \hat{i} + \frac{3}{4} \hat{j}) \times 4 = 2 \hat{i} + 3 \hat{j} m/s

Instantaneous power:

P_{in s t} = F \cdot v = 2 \times 2 + 3 \times 3 = 4 + 9 = 13 W

Ratio:

\frac{⟨ P ⟩}{P _{in s t}} = \frac{6}{13}

JEE Main · 2025 · 28 Jan 2025 · Shift 2 · Single correct option