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.