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 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.
