PROJECTILE MOTION

1. Introduction

Projectile motion is the motion of a body thrown into space with an initial velocity and then allowed to move freely under the action of gravity alone. Air resistance is neglected in basic projectile motion analysis.
This chapter is important for understanding the motion of bullets, balls, stones, shells, and many engineering applications.

2. Projectile

A projectile is any object that is given an initial velocity and thereafter moves under the influence of gravity only.

Examples:

• A stone thrown into the air
• A ball kicked at an angle
• A bullet fired from a gun

3. Assumptions in Projectile Motion

1. Air resistance is neglected
2. Acceleration due to gravity (g) is constant
3. Motion takes place near the earth’s surface
4. The earth’s curvature and rotation are ignored

4. Types of Projectile Motion

Projectile motion is classified into two main types:

1. Horizontal Projectile
2. Oblique (Inclined) Projectile

5. Horizontal Projectile Motion

Definition

When a body is projected horizontally from a certain height with an initial velocity, its motion is called horizontal projectile motion.

Motion Characteristics

• Horizontal motion → Uniform velocity
• Vertical motion → Uniformly accelerated motion due to gravity
• Path followed → Parabolic

Equations

Let:

• $u$u = initial velocity
• $h$h = height of projection
• $g$g = acceleration due to gravity

1. Time of flight

$$t = \sqrt{\frac{2h}{g}}$$

2. Horizontal range

$$R = u \times t$$

3. Vertical displacement

$$y = \frac{1}{2}gt^2$$

6. Oblique Projectile Motion

Definition

When a projectile is projected at an angle $\theta$θ with the horizontal, its motion is called oblique projectile motion.

Resolution of Velocity

Initial velocity $u$u is resolved into:

• Horizontal component:

$$u_x = u \cos \theta$$

• Vertical component:

$$u_y = u \sin \theta$$

7. Important Terms in Projectile Motion

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7.1 Time of Flight (T)

The total time the projectile remains in the air.$$T = \frac{2u \sin \theta}{g}$$

7.2 Maximum Height (H)

The highest vertical distance reached by the projectile.$$H = \frac{u^2 \sin^2 \theta}{2g}$$

7.3 Horizontal Range (R)

The horizontal distance covered by the projectile.$$R = \frac{u^2 \sin 2\theta}{g}$$

7.4 Angle for Maximum Range

Maximum range occurs when:$$\theta = 45^\circ$$

8. Trajectory of Projectile

• The path followed by a projectile is a parabola.
• Equation of trajectory:

$$y = x\tan\theta – \frac{g x^2}{2u^2 \cos^2\theta}$$

9. Velocity at Any Point

Velocity at any point during projectile motion is obtained by combining:

• Horizontal velocity (constant)
• Vertical velocity (changes due to gravity)

10. Special Cases

1. Projection at 0° → Horizontal motion
2. Projection at 90° → Vertical motion
3. Complementary angles (θ and 90° − θ) give the same range

11. Applications of Projectile Motion

• Ballistics and weapon design
• Sports (football, cricket, basketball)
• Water jets and fountains
• Motion of particles in engineering systems

12. Advantages of Projectile Analysis

• Helps predict motion path
• Useful in safety and design calculations
• Simplifies motion analysis by resolving components