Equation of a Trajectory

What is the trajectory of a projectile?

The trajectory of a projectile is the path it follows during its motion
The modelling assumptions mean it is symmetrical
It follows a path called a parabola
So far, motion has been described using the suvat formulae and we have produced parametric equations with time as the parameter
A Cartesian equation links the horizontal (x ) and vertical (y) components of the displacement $y = f (x)$

How do I find the equation for the trajectory of a projectile?

U is the initial speed of the projectile at an angle of θ° to the horizontal

2.6.3 Equation of a Trajectory Diagram 1_1

2.6.3 Equation of a Trajectory Diagram 1_2

Worked example

A particle is projected from a point on horizontal ground at a speed of V m s^-1 at an angle α° to the horizontal. The trajectory of the particle is given by the equation

$y = x \tan α - \frac{g x^{2}}{2 V^{2} \cos^{2} α}$

where $y$ is the vertical height of the particle and $x$ is the horizontal distance travelled by the particle.

(a) State two assumptions that have been made.

(b) In the case when V = 16 m s^-1 and α = 70° find, to two significant figures, the height of the particle after it has travelled 12 m horizontally.

(a) State two assumptions that have been made.

(b) In the case when V = 16 m s^-1 and α = 70° find, to two significant figures, the height of the particle after it has travelled 12 m horizontally.

2-6-3-equation-of-a-trajectory-example-solution-aqa

Examiner Tip

The steps are always the same so the only way the questions can be made difficult is for them to use confusing letters. They could use similar letters to suvat including capital letters. Alternatively, they could use unusual letters like a, instead of θ, for angles.
Make sure you show enough working out to convince the examiner that you know the steps especially if it is a "show that" question.
Sometimes you might have to use trigonometric identities from pure maths.

Equation of a Trajectory (AQA A Level Maths: Mechanics)

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1. Mechanics Toolkit

1.1 Quantities, Units & Modelling

1.1.1 Scalars & Vectors

1.1.2 Fundamental Units

1.1.3 Derived Units

1.1.4 Types of Force

1.1.5 Modelling Assumptions

1.2 Working with Vectors

1.2.1 Working with Vectors

2. Kinematics

2.1 Kinematics Graphs

2.1.1 Displacement-Time Graphs

2.1.2 Velocity-Time Graphs

2.1.3 Drawing Travel Graphs

2.2 Variable Acceleration - 1D

2.2.1 Using Calculus in 1D

2.3 Constant Acceleration - 1D

2.3.1 Deriving the suvat Equations

2.3.2 suvat in 1D

2.3.3 Acceleration due to Gravity

2.4 Variable Acceleration - 2D

2.4.1 Using Calculus in 2D

2.5 Constant Acceleration - 2D

2.5.1 suvat in 2D

2.6 Projectiles

2.6.1 Horizontal & Vertical Components

2.6.2 Using suvat

2.6.3 Equation of a Trajectory

2.6.4 Deriving Projectile Formulae

3. Forces & Newton's Laws

3.1 Forces

3.1.1 Force Diagrams

3.1.2 Equilibrium in 1D

3.1.3 Equilibrium in 2D

3.1.4 Forces in 2D - Vector Notation

3.2 Newton's Second Law

3.2.1 F = ma

3.2.2 Connected Bodies - Ropes & Tow Bars

3.2.3 Connected Bodies - The Lift Problem

3.2.4 Connected Bodies - Pulleys

3.2.5 F = ma - Vector Notation

3.3 Further Forces & Newton's Laws

3.3.1 Resolving Forces & Inclined Planes

3.3.2 Coefficient of Friction

3.3.3 Coefficient of Friction - F = ma

3.3.4 Coefficient of Friction - Inclined Planes

3.3.5 Coefficient of Friction - Harder Problems

4. Moments

4.1 Moments

4.1.1 Moments Diagrams

4.1.2 Using Moments - Equilibrium

4.1.3 Centres of Mass

4.1.4 Tilting

Author: Dan