PROJECTILE MOTION

PRERNA FOR IAS

PROJECTILE MOTION

1. Projectile Motion

Projectile motion is the motion of an object thrown into the air under the influence of gravity alone. It is a two-dimensional motion consisting of horizontal and vertical components. The horizontal velocity remains constant because there is no horizontal acceleration, while the vertical velocity changes due to gravity. The path followed by a projectile is called its trajectory, which is a parabola. Examples include a football kicked into the air, a bullet fired from a gun, or a stone thrown at an angle. Projectile motion helps us understand the relationship between displacement, velocity, acceleration, time, and gravity.

2. Equation of Trajectory

The trajectory of a projectile is the curved path followed by an object during its motion. By combining horizontal and vertical motion equations, the trajectory equation is obtained as:

y = x tanθ − (g x²)/(2u² cos²θ)

This equation shows that the path depends on the angle of projection, initial velocity, and gravitational acceleration. Since the equation is quadratic in x, the trajectory is a parabola. The projectile rises, reaches a maximum height, and then falls back to the ground. Understanding the trajectory helps in predicting the position of projectiles and is widely used in sports, engineering, military science, and space applications.

3. Time of Flight

Time of flight is the total time a projectile remains in the air before returning to the ground. It depends on the initial velocity and angle of projection. The formula for a projectile launched and landing at the same level is:

T = (2u sinθ)/g

where u is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity. Time of flight increases with greater vertical velocity. At the highest point, the vertical component of velocity becomes zero. This concept is important in sports, ballistics, and engineering for determining how long an object stays airborne.

4. Horizontal Range

Horizontal range is the maximum horizontal distance traveled by a projectile before it lands. It is given by:

R = (u² sin2θ)/g