Question

The displacement of a particle varies with time t, $x=a{e}^{-at}+b{e}^{\beta t}$ where a, b, $\alpha$ and $\beta$ are positive constants. The velocity of the particle will?

• A. Go on decreasing with time
• B. Be independent of $\alpha$ and $\beta .$
• C. Drop to zero when$\alpha =\beta$
• D. Go on increasing with time

Answer 1

Answer: (D)

According to the question

Displacement is given as $x=a{e}^{-\alpha t}+b{e}^{\beta t}$

We know that when we differentiate the displacement with respect to the time it gives velocity

So,

Velocity v = $\frac{dx}{dt}=\frac{d}{dt}\left(a{e}^{-\alpha t+b{e}^{\beta t}}\right)$

Or,
$=a{e}^{-\alpha t}(-\alpha )+b{e}^{\beta t}\left(\beta \right)=-a\alpha e^{-\alpha t}+b\beta e^{\beta t}$
From the above equation itself, we can tell that velocity will increase with time since the negative component will decrease with an increase in time and the positive component will increase in the equation given equation.