Question

Two unequal Masses P and Q moving along a straight line are brought to rest by applying equal retarding forces . If P moves twice the time as Q but goes only (1/3)^rd of the distance covered by Q before coming to rest , then find the ratio of their initial velocities .

Answer 1

Let the retarding force be F.

Suppose P having mass ‘m’ was initially moving with velocity ‘u’. It comes to rest in time ‘t’ after traveling a distance ‘x’.

Retardation of this mass is, a = F/m

And Q having mass ‘M’ was initially moving with velocity ‘U’. It comes to rest in time ‘T’ after traveling a distance ‘X’.

Retardation of this mass is, P = F/M

According to question,

t = 2T

x = X/3

Also,

F = ma = MA

For P,

0 = u – at

=> a = u/t

Similarly,

A = U/T

So,

a/A = uT/(Ut)

=> a/A = uT/(U × 2T)

=> a/A = u/(2U) ……………….(1)

Again for P,

0² = u² – 2ax

=> a = u²/(2x)

Similarly,

A = U²/(2X)

So,

a/A = [u²/(2x)]/[U²/(2X)]

=> a/A = (u²/U²)(X/x)

=> a/A = (u²/U²)[X/(X/3)]

=> a/A = 3(u²/U²) ………………(2)

(1) and (2) => u/(2U) = 3(u²/U²)

=> ½ = 3 u/U

=> u/U = 1/6

Thus, ratio of initial velocity of P to initial velocity of Q is = 1 : 6.



Hope it helps