Question

A particle moves on a straight line with a speed v that changes with time t according to the equation v = v0e–βt, where v0 and β are constants. The acceleration a of the particle is related to its speed by which of the following equations?

Answer 1

Given:

Velocity-Time relation :

$v = v_{0} {e}^{ - \beta t}$

To find:

Relation between velocity and acceleration?

Calculation:

$v = v_{0} {e}^{ - \beta t}$

• Differentiation with respect to time:

$\implies \: a = \dfrac{dv}{dt}$

$\implies a= \dfrac{d(v_{0} {e}^{ - \beta t} )}{dt}$

$\implies a= - v_{0} \beta {e}^{ - \beta t}$

• Replacing and putting value of v :

$\implies a= - \beta (v_{0} {e}^{ - \beta t} )$

$\boxed{ \implies a= - \beta v}$

So, this is the acceleration-velocity relation.