A high-performance motorboat starts from rest at a marker on a calm lake and accelerates in a straight line.
Using a simple model of constant acceleration a=2.4 m s−2a = 2.4 \text{ m s}^{-2}a=2.4 m s−2, a technician predicts that the velocity of the boat, exactly 5 seconds after starting from rest, is 12 m s−112 \text{ m s}^{-1}12 m s−1. Show how the technician obtained this prediction.
Using a refined model that accounts for water resistance, the boat's acceleration, a m s−2a \text{ m s}^{-2}a m s−2, at time ttt seconds after starting is given by the differential equation
dvdt=2.4−0.4v \frac{dv}{dt} = 2.4 - 0.4v dtdv=2.4−0.4vwhere v m s−1v \text{ m s}^{-1}v m s−1 is the velocity of the boat at time ttt. Find an expression for vvv in terms of ttt.
Compare the behavior of the velocity vvv as t→∞t \to \inftyt→∞ 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.
