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 AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) 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.
