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.
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
