The vertical velocity vvv of a test projectile, in m s−1\text{m s}^{-1}m s−1, is modelled by the function v(t)=t(21−t)v(t) = \sqrt{t}(21 - t)v(t)=t(21−t) for t≥0t \ge 0t≥0, where ttt is the time in seconds after launch.
Find, using calculus, the time ttt at which the projectile reaches its maximum vertical velocity.
A graph of vvv against ttt shows a region R1R_1R1 bounded by the curve and the ttt-axis between t=0t = 0t=0 and the point where the velocity first returns to zero (t=21t = 21t=21). A second region R2R_2R2 is bounded by the curve, the ttt-axis, and the vertical line t=Kt = Kt=K, where K>21K > 21K>21.
Given that the area of R1R_1R1 is equal to the area of R2R_2R2, use calculus to determine the exact value of KKK.
864 exam-style questions on Edexcel A Level Maths Integration, 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. Each one has a worked solution and a mark scheme showing where the marks go.