Projectile Motion (Level 2) - Physics

1. Fundamental Concepts

Definition: Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity.
Horizontal Motion: The horizontal component of projectile motion is constant velocity due to no horizontal forces (neglecting air resistance).
Vertical Motion: The vertical component of projectile motion is accelerated motion due to gravity, described by the equation $$y = y_0 + v_{0y}t - \frac{1}{2}gt^2$$.

2. Key Concepts

Range Equation: $$R = \frac{{v_0^2 \cdot \sin(2\theta)}}{g}$$

Time of Flight: $$T = \frac{{2v_0 \cdot \sin(\theta)}}{g}$$

Maximum Height: $$H = \frac{{v_0^2 \cdot \sin^2(\theta)}}{2g}$$

3. Examples

Example 1 (Basic)

Problem: A ball is thrown with an initial velocity of 20 m/s at an angle of 30 degrees. Find the range.

Step-by-Step Solution:

Identify given values: $$v_0 = 20 \text{ m/s}$$, $$\theta = 30^\circ$$, $$g = 9.8 \text{ m/s}^2$$
Substitute into the range formula: $$R = \frac{{20^2 \cdot \sin(2 \cdot 30)}}{9.8}$$
Calculate: $$R = \frac{{400 \cdot \sin(60)}}{9.8} = \frac{{400 \cdot 0.866}}{9.8} = 35.3 \text{ meters}$$

Validation: Substitute values → Original: $$R = \frac{{400 \cdot 0.866}}{9.8} = 35.3 \text{ meters}$$ ✓

Example 2 (Intermediate)

Problem: A projectile is launched from a height of 10 meters with an initial velocity of 30 m/s at an angle of 45 degrees. Find the total time of flight and the maximum height reached.

Step-by-Step Solution:

Identify given values: $$v_0 = 30 \text{ m/s}$$, $$\theta = 45^\circ$$, $$g = 9.8 \text{ m/s}^2$$, $$y_0 = 10 \text{ meters}$$
Calculate time of flight using the vertical component: $$T = \frac{{2v_0 \cdot \sin(\theta)}}{g} = \frac{{2 \cdot 30 \cdot \sin(45)}}{9.8} = \frac{{60 \cdot 0.707}}{9.8} = 4.3 \text{ seconds}$$
Calculate maximum height: $$H = \frac{{v_0^2 \cdot \sin^2(\theta)}}{2g} + y_0 = \frac{{30^2 \cdot \sin^2(45)}}{2 \cdot 9.8} + 10 = \frac{{900 \cdot 0.5}}{19.6} + 10 = 25.3 \text{ meters}$$

Validation: Substitute values → Original: $$T = 4.3 \text{ seconds}$$, $$H = 25.3 \text{ meters}$$ ✓

4. Problem-Solving Techniques

Decomposition Strategy: Break down the problem into horizontal and vertical components.
Graphical Representation: Use graphs to visualize the trajectory and understand the motion better.
Unit Consistency: Ensure all units are consistent before performing calculations.