Question

Two particles P and Q start from rest and move for equal time on a straight line. Particle P has an acceleration of$Xm/s^2$ for the first half of the total time and $2xm/s^2$ for the second half. Particle Q has an acceleration of $2Xm/s^2$ for the first half of the total time and $Xm/s^2$ for the second half. Which particle has covered larger distance?

• A. Both have covered the same distance
• B. P has covered the larger distance
• C. Q has covered the larger distance
• D. data insufficient

Answer 1

The correct option is C Q has covered the larger distance

For first half acceleration

Particle $P=X㎨$

Particle $Q=2X㎨$

For second half acceleration

Particle $P=2X㎨$

Particle $Q=X㎨$

For particle P

Let the total time for complete journey be T

For first half

Time$=\frac{T}{2}$sec and acceleration$=X㎨$

$\therefore S_1=ut+\frac{1}{2}\times {\left(\frac{T}{2}\right)}^2$

Initial velocity$=u=0$

$\therefore S_1=\frac{X{T}^2}{8}\to 1$

Now particle has gained some velocity

$v=u+\frac{XT}{2}$

$\therefore v=\frac{XT}{2}-(\because u=0)$

For second half

$S_2=ut+\frac{1}{2}\left(2X\right){\left(\frac{T}{2}\right)}^2$

Put $u=\frac{XT}{2}$ as first half final velocity becomes initial velocity and $t=\frac{T}{2}$

$\therefore S_2=\frac{XT}{2}\times \frac{T}{2}+\frac{1}{2}\left(2X\right)\frac{T^2}{4}$

$\therefore S_2=\frac{X{T}^2}{2}\to 2$

Total distance moved by particle P

$S=S_1+S_2=\frac{X{T}^2}{8}+\frac{X{T}^2}{2}=\frac{5X{T}^2}{8}$

For particle Q, for first half

$S_1=ut+\frac{2X{t}^2}{2}(\because u=0)$

$S_2=\frac{X{T}^2}{4}\to 3$

Now the particle is having velocity

$v=u+2\times \frac{T}{2}=XT$

For second half $S_2=ut+\frac{1}{2}\times t^2$

(Put $u=XT,t=\frac{T}{2}$)

$S_2=XT\times \frac{T}{2}+\frac{1}{2}\times \frac{T^2}{4}=\frac{5X{T}^2}{8}$

Total distance moved by particle Q

$S=S_1+S_2$

$=\frac{X{T}^2}{4}+\frac{5X{T}^2}{8}$

$=\frac{7X{T}^2}{8}$

On comparing total distance covered by P and Q i.e.,$\frac{5X{T}^2}{8}$ and $\frac{7X{T}^2}{8}$ respectively.

It is seen that particle Q has covered the larger distance