Short Answer:

Projectile motion is the motion of a body that is thrown or projected into the air and moves under the influence of gravity alone, without being powered by any engine or force after it is projected. The path followed by the body is called a trajectory, and it is usually parabolic in shape.

In projectile motion, two motions occur simultaneously — a horizontal motion with constant velocity and a vertical motion with constant acceleration due to gravity. Common examples include a thrown ball, a bullet fired from a gun, or water coming out of a fountain nozzle.

Detailed Explanation :

Projectile Motion

Projectile motion is a special type of motion in mechanics where a body is thrown into space and then moves freely under the influence of gravity. After projection, the only force acting on the body is gravity, which pulls it downward. Air resistance is usually neglected for simplicity. The motion occurs in a two-dimensional plane, meaning the body moves horizontally as well as vertically at the same time.

The path that the projectile follows during its flight is known as the trajectory, and for most projectiles (except those launched vertically), this path is a parabola.

Some real-life examples of projectile motion include:

• A ball thrown by hand.
• A bullet fired from a gun.
• A stone thrown at an angle from the ground.
• Water flowing out of a hose pipe.
• A football kicked at an angle.

Components of Projectile Motion

Projectile motion can be divided into two parts for easier analysis:

1. Horizontal Motion:
   The projectile moves with constant velocity in the horizontal direction because there is no horizontal acceleration (ignoring air resistance). The horizontal distance covered is called the range of the projectile.
   The horizontal velocity component is constant and given by:

where,
= initial velocity of projection,
= angle of projection.

1. Vertical Motion:
   The projectile moves under the influence of gravity in the vertical direction. This motion is uniformly accelerated because gravity causes a constant downward acceleration .
   The vertical velocity component is given by:

As the projectile rises, its vertical velocity decreases due to gravity, and at the highest point, the vertical velocity becomes zero. Then it starts to fall downward, gaining velocity due to gravity again.

Equations of Motion for Projectile

For the projectile, both horizontal and vertical motions occur simultaneously but independently. The combined motion gives a curved path.

If a body is projected with an initial velocity  making an angle  with the horizontal, then:

• Horizontal displacement:

• Vertical displacement:

Eliminating  between these two equations gives the equation of the trajectory:

This is the equation of a parabola, proving that the path of a projectile is parabolic.

Important Terms in Projectile Motion

1. Time of Flight (T):
   It is the total time the projectile remains in the air before returning to the ground.

1. Maximum Height (H):
   It is the highest vertical distance reached by the projectile from the point of projection.

1. Horizontal Range (R):
   It is the total horizontal distance covered by the projectile before it hits the ground.

The range is maximum when .

Nature of the Trajectory

The trajectory of the projectile is a parabolic curve. This happens because:

• The horizontal motion has constant velocity (no horizontal acceleration).
• The vertical motion has uniform acceleration due to gravity.

The combination of these two independent motions results in a curved path — the parabola.

At the topmost point, the projectile has only horizontal velocity because its vertical velocity becomes zero at that instant.

Examples in Daily Life

1. Throwing a ball at an angle.
2. A cannon firing a shell.
3. Water thrown upward at an angle from a hose pipe.
4. A javelin thrown by an athlete.
5. A football kicked during a game.

In all these cases, the objects follow a curved path and are affected only by gravity once released.

Graphical Representation

If we draw a velocity-time graph:

• The horizontal velocity remains constant, forming a straight line parallel to the time axis.
• The vertical velocity decreases linearly with time until it becomes zero at the highest point, then increases in the downward direction due to gravity.

The combination of these motions explains the curved (parabolic) trajectory.

Conclusion

Projectile motion is the motion of a body that is thrown into space and moves under the influence of gravity only. It involves two simultaneous motions — horizontal motion with uniform velocity and vertical motion with uniform acceleration. The path followed by the projectile is parabolic in nature. Understanding projectile motion is very important in mechanical engineering and physics because it helps in analyzing the motion of projectiles, machines, missiles, and various other engineering systems involving trajectories.