How do you find the time of flight of a projectile?

Set y=0 and take the non-zero root.

How do you find the range?

Substitute the time of flight into x.

How do you find the greatest height?

Use symmetry (t halfway) or \frac{dy}{dt}=0, then find y.

Convert x=80\cos\theta, y=50\sin\theta to Cartesian form.

\dfrac{x^2}{6400}+\dfrac{y^2}{2500}=1.

When do two objects collide?

When they are at the same point at the same time.

Paths cross at a point. Does this prove a collision?

No; the objects must reach it at the same time.

Name a limitation of a projectile model.

It ignores air resistance (and spin and the object's size).

What should you state before using a model?

Meaning of variables, units, and the valid range of t.

Parametric equations in modellingAQA A-Level Maths: Flashcards

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Why use parametric equations in modelling?

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Why use parametric equations in modelling?: They give position $(x,y)$ and the time $t$ together.
How do you find the time of flight of a projectile?: Set $y=0$ and take the non-zero root.
How do you find the range?: Substitute the time of flight into $x$ .
How do you find the greatest height?: Use symmetry ( $t$ halfway) or $\frac{dy}{dt}=0$ , then find $y$ .
Model for a wheel: $x=r\sin\omega t$ , $y=h-r\cos\omega t$ . Period?: $\dfrac{2\pi}{\omega}$ .
Greatest and least $y$ for $y=h-r\cos\omega t$ ?: $h+r$ and $h-r$ .
Convert $x=80\cos\theta$ , $y=50\sin\theta$ to Cartesian form.: $\dfrac{x^2}{6400}+\dfrac{y^2}{2500}=1$ .
When do two objects collide?: When they are at the same point at the same time.
Paths cross at a point. Does this prove a collision?: No; the objects must reach it at the same time.
Name a limitation of a projectile model.: It ignores air resistance (and spin and the object's size).
Distance between $(25,10)$ and $(30,8)$ ?: $\sqrt{29}\approx5.4$ .
What should you state before using a model?: Meaning of variables, units, and the valid range of $t$ .
Ball: $y=16t-5t^2$ . Time of flight?: $3.2$ s.

Exam questions on Parametric equations in modelling

A ball is kicked from level ground. At time $t$ seconds after the kick, its horizontal distance from the starting point is $x=12t$ metres and its height is $y=16t-5t^2$ metres. The model applies until the ball lands.
Find the greatest height reached by the ball.2 marks
A Ferris wheel has its centre $12$ m above the ground. The position of a passenger pod is modelled by $x=10\sin(0.2t)$ , $y=12-10\cos(0.2t)$ , where $x$ is the horizontal distance in metres from the vertical line through the centre, $y$ is the height above the ground in metres and $t$ is the time in seconds after the pod passes its lowest point. The angle $0.2t$ is in radians.
State the greatest height of the pod above the ground, and justify your answer.2 marks
A runner follows an oval track. Relative to the centre of the track, the runner's position at time $t$ seconds is $x=80\cos\theta$ , $y=50\sin\theta$ , where $\theta=0.1t$ radians and $x$ and $y$ are in metres.
Show that the runner's path has the Cartesian equation $\dfrac{x^2}{6400}+\dfrac{y^2}{2500}=1$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).