CAIE A-Level Further Mathematics 3.1.3 Projectile Trajectories

Practise eliminating time to obtain the Cartesian trajectory, then applying point, height or obstacle conditions to solve for projection speed, angle or distance.

Exam points

eliminate time from the component equations and retain the squared horizontal-speed factor
substitute the stated point, height or obstacle into the trajectory before solving the unknown parameter
when two paths are possible, solve the resulting quadratic and report both admissible projection angles

[Maximum number: 2]

A particle P is projected with speed $u \mathrm{~ms}^{-1}$ at an angle $\theta$ above the horizontal from a point O on a horizontal plane and moves freely under gravity. During its flight P passes through the point which is a horizontal distance 3 a from O and a vertical distance $\frac{3}{8} a$ above the horizontal plane. It is given that $\tan \theta=\frac{1}{3}$ .

Show that $u^{2}=8 a g$ .
A particle Q is projected with speed $V \mathrm{~ms}^{-1}$ at an angle $\alpha$ above the horizontal from O at the instant when P is at its highest point. Particles P and Q both land at the same point on the horizontal plane at the same time.

Use equation of trajectory: $y=x \tan \alpha-\frac{g x^{2}}{2 u^{2}} \sec ^{2} \alpha$ :
$\frac{3}{8} a=3 a \times \frac{1}{3}-g \times \frac{(3 a)^{2}}{2 u^{2}}\left(1+\frac{1}{9}\right)$

M1

$\frac{5}{8} a=\frac{5 g a}{u^{2}}, \quad u^{2}=8 g a$

A1

At least one step of working.
AG

2

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3.1.3—Projectile motion question 1