Question

The displacement $x$ of a particle varies with time $t , x = a e ^ { - \alpha t } + b e ^ { \beta t }$ , 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)

Go on increasing with time

Answer 2

$x = a e ^ { - \alpha t } + b e ^ { \beta t }$

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

$\left. = a \cdot e ^ { - \alpha t } ( - \alpha ) + b e ^ { \beta t } \cdot \beta \right)$ $= - a \alpha e ^ { - \alpha t } + b \beta e ^ { \beta t }$

Acceleration $= - a \alpha e ^ { - \alpha t } ( - \alpha ) + b \beta e ^ { b t } . \beta$

$= a \alpha ^ { 2 } e ^ { - \alpha t } + b \beta ^ { 2 } e ^ { \beta t }$

Acceleration is positive so velocity goes on increasing with time.