A small object is dropped from a high altitude and falls vertically towards the ground.
Using a simple model of constant acceleration, a student predicts that the velocity of the object, exactly 5 seconds after being released from rest, is 5g m s−15g\text{ m s}^{-1}5g m s−1. Show how the student has obtained this prediction.
Using a refined model that accounts for air resistance, it is assumed that the object's acceleration, a m s−2a\text{ m s}^{-2}a m s−2, at time ttt seconds after release is given by
a=g−0.2v a = g - 0.2v a=g−0.2vwhere v m s−1v\text{ m s}^{-1}v m s−1 is the velocity of the object at time ttt. Find an expression for vvv in terms of ttt.
Compare the behavior of vvv as t→∞t \to \inftyt→∞ for both models.
480 exam-style questions on AQA A Level Maths 1.11 H: Integration, 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). Each one has a worked solution and a mark scheme showing where the marks go.