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.
363 exam-style questions on CCEA A Level Maths 3.6 Integration (A-level only), covering 3.6.1 Integration (A-level only), 3.6.2 Integration (A-level only), 3.6.3 Integration (A-level only), 3.6.4 Integration (A-level only), 3.6.5 Integration (A-level only), 3.6.6 Integration (A-level only), and 3.6.7 Integration (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.