Question

A point moves rectilinearly.Its displacement $x$ at time $t$ is given by $x^2=t^2+1$. Its acceleration at time $t$ is

• A. $\frac{1}{x^3}$
• B. $\frac{1}{x}-\frac{1}{x^2}$
• C. $\frac{1}{x^2}$
• D. $\frac{t^2}{x^3}$

Answer 1

The correct option is A $\frac{1}{x^3}$

Displacement equation is given by

$x^2=t^2+1$ . . . . . . . . .(1)

$x=(t^2+1{)}^{1/2}$

Velocity, $v=\frac{dx}{dt}=t(1+t^2{)}^{-1/2}$

Acceleration, $a=\frac{dv}{dt}$

$a=\frac{1}{(t^2+1{)}^{1/2}}-\frac{t^2}{(t^2+1{)}^{3/2}}$

$a=\frac{1}{x}-\frac{(x^2-1)}{x^3}$ (from equation 1)

$a=\frac{x^2-x^2+1}{x^3}=\frac{1}{x^3}$

The acceleration at time $t$ is $\frac{1}{x^3}$.

The correct option is A.