A vertical-launch research drone is programmed to reach a target height of 500 metres. The drone’s vertical velocity, w m s−1w\text{ m s}^{-1}w m s−1, is modelled by the equation:
w=25.6−20e−0.5t−0.04e0.25t w = 25.6 - 20e^{-0.5t} - 0.04e^{0.25t} w=25.6−20e−0.5t−0.04e0.25twhere ttt is the time in seconds after the start of the ascent phase.
Find the maximum vertical velocity of the drone, giving your answer to one decimal place. Fully justify that the value you have found is a maximum.
Find an expression for the height of the drone above its starting position, hhh, in terms of ttt.
In a specific test flight, the drone reached the 500-metre target in exactly 28.0 seconds. Comment on the accuracy of the model.
