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.
438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.