Question

Two particles move in the same straight line starting at the same moment from the same point in the same direction. The first moves with constant velocity u and the second starts from rest with constant acceleration f. Then

• A. They will be at the greatest distance at the end of time \dfrac {u}{2f}$ from the start
• B. They will be at the greatest distance at the end of time \dfrac {u}{f}$ from the start
• C. Their greatest distance is $\frac{u^2}{2f}$
• D. Their greatest distance is $\frac{u^2}{f}$

Answer 1

Answer: (B), (C)

Their greatest distance is $\frac{u^2}{2f}$
They will be at the greatest distance at the end of time \dfrac {u}{f}$fromthestartLet$S$bethedistancebetweentheparticlesattime$t$

Particle 1 is $x_1=u$

Particle 2 is $x_2=\frac{1}{2}f{t}^2$

Then, $S=x_1-x_2$

$=ut-\frac{1}{2}f{t}^2$ ..... $\left(i\right)$

For $S$ being maximum

$\frac{dS}{dt}=0⟹u-ft=0$

$⟹t=\frac{u}{f}$

Substituting $t$ in $\left(i\right)$ we get

$S=u\frac{u}{f}-\frac{1}{2}f{\left(\frac{u}{f}\right)}^2$

$=\frac{u^2}{2f}$

Hence, they will be at the greatest distance and greatest distance is $\frac{u^2}{2f}$