### Introduction

In this Section we study the motion of projectiles constrained only by gravity. Although historically the mechanics of projectile motion were studied and developed mainly in military contexts, there are many relevant non-military situations. For example botanists study the mechanics of dispersal of seeds from ÔexplodingÕ pods; hydraulic engineers are interested in the distribution and settling of sediments and particles; many athletic activities and sports such as skiing and diving involve humans acting as projectiles through leaping or hurdling or otherwise throwing themselves about. Other sporting activities involve inanimate projectiles e.g. balls of various kinds, javelins. Precise models of some possible situations, for example swerving or swinging or spinning balls, or ski-jumping involve rather complicated kinds of motion and require considerations of resistive forces and aerodynamic forces. First trips around the modelling cycle (see
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HELM booklet
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5), sometimes second trips, are given here.

#### Prerequisites

- be able to use vectors and to carry out scalar and vector products
- be able to use Newton’s laws to describe and model the motion of particles
- be able to use coordinate geometry to study circles and parabolas
- be able to use calculus to differentiate and integrate polynomials

#### Learning Outcomes

- use vector notation to represent the position, velocity and acceleration of projectiles, objects moving on inclined planes and objects moving on curved paths
- compute frictional forces on static and moving objects on inclined planes and on objects moving at constant speed around bends

#### Contents

1 Introduction2 Projectiles: an introduction

2.1 Vertical motion under gravity

2.2 Velocity and acceleration vectors

3 Projectiles

3.1 Angled launches

3.2 Choosing trajectories

4 Energy and projectile motion

5 Projectiles on inclined planes