Projectile Motion

Definition and Path of a Projectile

Projectile motion refers to the motion of an object that is projected into the air and moves under the influence of gravity alone, following a curved trajectory. This type of motion is fundamental in physics and is commonly observed in sports, ballistics, and various engineering applications.

• Projectile: An object moving in two dimensions under the influence of Earth's gravity.

• Path: The trajectory of a projectile is a parabola.

• Examples: A thrown ball, a kicked football, or a leaping animal.

Independence of Horizontal and Vertical Motions

Projectile motion can be analyzed by separating it into horizontal and vertical components. Each component behaves independently, except for sharing the same time of flight.

• Horizontal Motion: The horizontal velocity () remains constant throughout the flight, as there is no acceleration in the x-direction (assuming air resistance is negligible).

• Vertical Motion: The vertical velocity () changes due to the constant acceleration of gravity ().

• Key Point: The size of the velocity arrows in the x-direction remains constant, while those in the y-direction change due to gravity.

Kinematic Equations for Projectile Motion

The following equations describe the position and velocity of a projectile at any time t:

• Horizontal Motion (no acceleration):

• Vertical Motion (constant acceleration):

• Vertical velocity squared:

Note: is the acceleration due to gravity ( downward).

Projectile Launched at an Angle

When a projectile is launched at an initial angle to the horizontal, its initial velocity has both horizontal and vertical components:

• Horizontal component:

• Vertical component:

The analysis proceeds by applying the kinematic equations to each component separately.

Demonstration: Independence of Motions

A classic demonstration involves dropping two balls from the same height: one with an initial horizontal velocity and one with none. Both hit the ground at the same time, showing that horizontal and vertical motions are independent.

• Observation: The vertical positions of both balls are identical at identical times, while the horizontal position of the ball with initial velocity increases linearly.

Summary Table: Key Properties of Projectile Motion

Example Applications

• Horizontal Launch: A tiger leaps horizontally from a 6.5-m high rock with a speed of 3.5 m/s. To find the distance from the base where she lands:

  1. Calculate the time to fall:

  2. Calculate horizontal distance:

• Angled Launch: A football is kicked at 18.0 m/s at 35.0° to the horizontal.

  1. Find vertical component:

  2. Time to reach maximum height:

  3. Total time of flight:

  4. Horizontal distance:

Parabolic Nature of Projectile Motion

By eliminating time from the kinematic equations, the trajectory equation is obtained:

This equation is quadratic in , confirming the path is a parabola.

Additional info:

• Projectile motion problems often require finding time of flight, maximum height, and range.

• Air resistance is neglected in introductory analysis.