Projectile Motion

Have you ever watched a cricket ball soaring through the air after being hit by a batsman, or noticed the curved path of water from a hose? These are examples of projectile motion -- the motion of an object launched into the air that moves under the influence of gravity alone.

What is Projectile Motion?

A projectile is any object thrown or launched into the air with an initial velocity, after which it moves only under gravity (ignoring air resistance). The path it follows is called a trajectory, and it is always a parabola.

Breaking Down the Motion

The key insight is to treat horizontal and vertical motions independently:

Horizontal motion: No acceleration (ax = 0), so horizontal velocity remains constant.

• x = ux t (where ux = u cos theta)

Vertical motion: Acceleration due to gravity (ay = g = 9.8 m/s² downward).

• vy = uy - gt
• y = uy t - (1/2)g t²

Key Formulas

Time of flight: T = 2u sin theta / g

Maximum height: H = u² sin² theta / (2g)

Range: R = u² sin 2theta / g

Maximum range occurs when theta = 45 degrees.

Worked Example

A ball is thrown at 20 m/s at an angle of 30 degrees to the horizontal. Find the time of flight, maximum height, and range. (Take g = 10 m/s²)

Step 1: Identify components.

• ux = 20 cos 30 = 20 x 0.866 = 17.32 m/s
• uy = 20 sin 30 = 20 x 0.5 = 10 m/s

Step 2: Time of flight.

• T = 2 x 20 x sin 30 / 10 = 2 x 20 x 0.5 / 10 = 2 seconds

Step 3: Maximum height.

• H = (20)² x (sin 30)² / (2 x 10) = 400 x 0.25 / 20 = 5 metres

Step 4: Range.

• R = (20)² x sin 60 / 10 = 400 x 0.866 / 10 = 34.64 metres

Real-World Applications

• In Nepal, soldiers at mountainous border posts need projectile calculations for artillery
• Athletes in javelin, shot put, and discus throw optimize their launch angle
• Water fountains in Kathmandu's gardens use projectile principles

Key Takeaways

• Projectile motion is a combination of uniform horizontal motion and uniformly accelerated vertical motion
• The trajectory is always parabolic
• Range is maximum at a 45-degree launch angle
• Time of flight depends only on the vertical component of velocity