Projectile Motion (Level 1) - 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.

2. Key Concepts

Equations of Motion: $$x = v_0 \cdot \cos(\theta) \cdot t$$ $$y = v_0 \cdot \sin(\theta) \cdot t - \frac{1}{2} g t^2$$

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

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

3. Examples

Example 1 (Basic)

Problem: A ball is thrown with an initial velocity of $$5 \text{ m/s}$$ at an angle of $$45^\circ$$. Find the range.

Step-by-Step Solution:

Substitute values into the range formula: $$R = \frac{{5^2 \cdot \sin(2 \cdot 45^\circ)}}{9.81}$$
Simplify: $$R = \frac{25 \cdot \sin(90^\circ)}{9.81}$$
Calculate: $$R = \frac{25 \cdot 1}{9.81} \approx 2.55 \text{ m}$$

Validation: Substitute values → Original: $$R = \frac{25 \cdot 1}{9.81} \approx 2.55 \text{ m}$$; Simplified: $$2.55 \text{ m}$$ ✓

Example 2 (Intermediate)

Problem: A projectile is launched from a height of $$10 \text{ m}$$ with an initial velocity of $$20 \text{ m/s}$$ at an angle of $$30^\circ$$. Find the total time of flight.

Step-by-Step Solution:

Use the vertical motion equation: $$y = 20 \cdot \sin(30^\circ) \cdot t - \frac{1}{2} \cdot 9.81 \cdot t^2 + 10$$
Simplify: $$y = 10 \cdot t - 4.905 \cdot t^2 + 10$$
Set y = 0 and solve for t: $$0 = 10 \cdot t - 4.905 \cdot t^2 + 10$$
Quadratic formula: $$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a = -4.905$$, $$b = 10$$, and $$c = 10$$
Calculate: $$t = \frac{-10 \pm \sqrt{10^2 - 4(-4.905)(10)}}{2(-4.905)}$$
Positive root: $$t \approx 3.43 \text{ s}$$

Validation: Substitute values → Original: $$t = \frac{-10 \pm \sqrt{10^2 - 4(-4.905)(10)}}{2(-4.905)}$$; Simplified: $$t \approx 3.43 \text{ s}$$ ✓

4. Problem-Solving Techniques

Separate Components: Always separate the problem into horizontal and vertical components.
Use Diagrams: Draw diagrams to visualize the motion and identify angles and distances.
Check Units: Ensure all units are consistent before solving equations.