An autonomous underwater vehicle (AUV) is released from the surface of the ocean and descends vertically towards the seabed.
Using a simple model of constant acceleration 0.25 m s-2, a researcher predicts that the velocity of the AUV exactly 80 seconds after being released from rest is 20 m s-1. Show how the researcher has obtained this prediction.
Using a refined model that accounts for fluid resistance, it is assumed that the AUV's acceleration, a m s−2a \text{ m s}^{-2}a m s−2, at time t t\,t seconds after release is given by
a=0.25−0.01v a = 0.25 - 0.01v a=0.25−0.01vwhere v m s−1v \text{ m s}^{-1}v m s−1 is the velocity of the AUV at time ttt. Find an expression for v v\,v in terms of ttt.
Compare the behavior of v v\,v as t→∞ t \to \infty\,t→∞ for both models.
50 exam-style questions on Edexcel A Level Maths 11.11 Modelling with Differential Equations. Each one has a worked solution and a mark scheme showing where the marks go.