Question

Mass of 2 bodies are in ratio 1:2. Both are initially at rest and are subjected to same force. Velocity acquired by them after covering a fixed distance ae in ratio

(a) 1:1
(b) 1:2
(c) 1:root2
(d) root2:1

Answer 1

mass of the bodies are in ratio 1:2
if same force is applied, the ratio of accelerations would be 2:1 since acceleration is inversely proportional to mass.

Let the acceleration be 2a and a.
initial velocities of both are zero, so u = 0 for both.
let the velocities acquired after covering distance d be V and v.

d = (V² - u²)/2*2a = (v² - u²)/2*a
⇒ (V² - 0)/4a = (v² - 0)/2a
⇒ V²/4a = v²/2a
⇒ V²/v² = 4a/2a = 2
⇒ V/v = √2
⇒ V:v = √2:1
Answer is D.

Answer 2

F = m a = constant
Masses are in the ratio:   1  :  2
So accelerations are in the ratio:  2  :  1

v² = u² + 2 a s
distance s = constant.   u = 0 in both cases.
v = √(2 a s)

velocity is proportional to square root of acceleration.  =>  √2 : 1