A golf ball is struck from a point O O\,O on level ground. Its position t t\,t seconds after being struck is modelled by
x=20tcosαx = 20t\cos\alphax=20tcosα, y=20tsinα−4.9t2y = 20t\sin\alpha - 4.9t^2y=20tsinα−4.9t2
where x x\,x metres and y y\,y metres are the horizontal and vertical displacements from OOO, and α \alpha\,α is the angle of projection above the horizontal.
Show that a Cartesian equation for the path of the ball is
y=xtanα−49x2sec2α4000\displaystyle y = x\tan\alpha - \frac{49x^2\sec^2\alpha}{4000}y=xtanα−400049x2sec2α
Given that α=30°\alpha = 30°α=30°, find the horizontal distance travelled by the ball before it first returns to the ground. Give your answer to three significant figures.
State one modelling assumption that has been made.
143 exam-style questions on OCR A Level Maths 1.3 Coordinate Geometry in the x-y Plane, covering 1.3.1 Straight lines, 1.3.2 Parallel and perpendicular lines, 1.3.3 Straight line models, 1.3.4 Coordinate geometry of a circle, 1.3.5 Centre and radius by completing the square, 1.3.6 Circle properties, 1.3.7 Parametric equations of curves (A-level only), 1.3.8 Parametric equations in context (A-level only), and 1.3 Coordinate Geometry in the x-y Plane. Each one has a worked solution and a mark scheme showing where the marks go.