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.
178 exam-style questions on OCR (MEI) A Level Maths 1.5 Coordinate Geometry, covering 1.5.1 Equation of a straight line y = mx + c, 1.5.2 Gradients of parallel and perpendicular lines, 1.5.3 Distance between two points, 1.5.4 Midpoint of a line segment, 1.5.5 Form the equation of a straight line, 1.5.6 Draw a line given its equation, 1.5.7 Point of intersection of two lines, 1.5.8 Straight line models, 1.5.9 Intersection of a line and a curve, 1.5.10 Intersection of a line and a circle, 1.5.11 Equation of a circle, 1.5.12 Circle properties, 1.5.13 Parameter and parametric equations (A-level only), 1.5.14 Convert between cartesian and parametric forms (A-level only), 1.5.15 Equation of a circle in parametric form (A-level only), 1.5.16 Gradient of a parametric curve (A-level only), 1.5.17 Parametric equations in modelling (A-level only), and 1.5 Coordinate Geometry. Each one has a worked solution and a mark scheme showing where the marks go.