A particle’s displacement, r metres, with respect to time, t seconds, is defined by the equation $r = 3e^{0.5t}$ Find an expression for the velocity, v m s$^{-1}$, of the particle at time t seconds - AQA - A-Level Maths Pure - Question 11 - 2021 - Paper 2

To find the velocity of the particle, we need to take the derivative of the displacement function with respect to time.

Given the displacement: $r = 3e^{0.5t}$

we calculate the velocity as follows:

$v = \frac{dr}{dt}$

Using the chain rule, we get:

$v = 3 \cdot \frac{d}{dt}(e^{0.5t})$

The derivative of $e^{kt}$ is $ke^{kt}$, where $k = 0.5$:

$\frac{d}{dt}(e^{0.5t}) = 0.5 e^{0.5t}$

Thus, substituting back, we have:

$v = 3 \cdot 0.5 e^{0.5t} = 1.5 e^{0.5t}$

Therefore, the expression for the velocity of the particle at time t seconds is:

$v = 1.5 e^{0.5t}$

This is the correct answer.