Mastering Motion: A Comprehensive Guide to Projectile Motion and Relative Velocity

1. Projectile Motion

Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory. We can analyze projectile motion by considering the horizontal and vertical components of its motion independently.

1.1 Horizontal Projectile Motion

In the absence of air resistance, the horizontal velocity of a projectile remains constant. This is because there is no horizontal force acting on the object once it has been projected. Thus, the horizontal motion is uniform motion.

Horizontal distance (x) = Initial horizontal velocity (vₓ) × Time (t)

1.2 Vertical Projectile Motion

The vertical motion of a projectile is influenced by gravity, which causes a constant downward acceleration (g ≈ 9.8 m/s²). This is uniformly accelerated motion.

Vertical displacement (y) = Initial vertical velocity (v_y) × Time (t) + 0.5 × g × t²
Final vertical velocity (v_y_f) = Initial vertical velocity (v_y_i) + g × t
v_y_f² = v_y_i² + 2 × g × y

Key Concepts for Projectile Motion:

• Time of Flight (T): The total time the projectile remains in the air.
• Maximum Height (H): The highest vertical position reached by the projectile.
• Range (R): The total horizontal distance traveled by the projectile.

Example: A ball is kicked horizontally from a cliff 20m high with an initial velocity of 15 m/s. Find the time it takes to reach the ground and its horizontal range.

Solution:

• Vertical motion: y = v_y_i*t + 0.5*g*t². Here, v_y_i = 0, y = 20m. So, 20 = 0.5 * 9.8 * t². Solving for t, t = sqrt(20 / 4.9) ≈ 2.02 seconds.
• Horizontal motion: x = v_x * t. Here, v_x = 15 m/s, t ≈ 2.02 s. So, x = 15 * 2.02 = 30.3 meters.

2. Relative Velocity

Relative velocity is the velocity of an object or observer with respect to another object or observer. It is crucial when dealing with motion in two or three dimensions, especially when the reference frame is also in motion.

V_AB = V_A - V_B

Where V_AB is the velocity of A relative to B, V_A is the velocity of A with respect to the ground, and V_B is the velocity of B with respect to the ground. This is a vector subtraction.

2.1 River Boat Problem

A classic application of relative velocity involves a boat moving in a river. The velocity of the boat with respect to the ground is the vector sum of its velocity with respect to the water and the velocity of the water (river current) with respect to the ground.

V_boat_ground = V_boat_water + V_water_ground

Scenario 1: Crossing the river directly (shortest path)
To cross directly, the boat must aim upstream to counteract the river current. The resultant velocity will be perpendicular to the river flow.

Scenario 2: Crossing in shortest time
To cross in the shortest time, the boat should head directly across the river. The time taken depends only on the boat's speed relative to the water perpendicular to the current and the width of the river.

2.2 Aeroplane Air Problem

Similar to the river boat problem, an aeroplane's velocity with respect to the ground is affected by the wind's velocity. The aeroplane's velocity relative to the air (airspeed) combined with the wind's velocity relative to the ground determines its velocity relative to the ground (groundspeed).

V_plane_ground = V_plane_air + V_air_ground

3. Graphs of Motion

Graphs provide a powerful way to visualize and analyze motion. We typically use displacement-time, velocity-time, and acceleration-time graphs.

3.1 Displacement-Time Graph (x-t graph)

• Slope: Represents velocity. A straight line indicates constant velocity. A curved line indicates changing velocity (acceleration).
• Horizontal line: Object is at rest.
• Steeper slope: Greater speed.

3.2 Velocity-Time Graph (v-t graph)

• Slope: Represents acceleration. A straight line with non-zero slope indicates constant acceleration.
• Area under the curve: Represents displacement.
• Horizontal line: Constant velocity (zero acceleration).
• Slope of zero: Constant velocity.

3.3 Acceleration-Time Graph (a-t graph)

• Area under the curve: Represents the change in velocity.
• Horizontal line: Constant acceleration.
• Line on x-axis: Zero acceleration (constant velocity).