Section 1.6: Projectile Motion
Projectile motion describes the motion of an object launched into the air subject to gravity. The motion can be analyzed as two independent components: horizontal and vertical.
Key Concepts:
- Horizontal motion: constant velocity \(v_x = v_0 \cos\theta\)
- Vertical motion: constant acceleration \(a_y = -g\), velocity \(v_y = v_0 \sin\theta - g t\)
- Time of flight: \(T = \frac{2 v_0 \sin\theta}{g}\)
- Maximum height: \(H = \frac{(v_0 \sin\theta)^2}{2g}\)
- Range: \(R = \frac{v_0^2 \sin 2\theta}{g}\)
Example 1
A ball is thrown with initial speed 20 m/s at 30° above the horizontal. Find the time of flight, maximum height, and horizontal range.
Time of flight: \(T = \frac{2 \cdot 20 \cdot \sin 30^\circ}{9.8} \approx 2.04 \text{ s}\)
Maximum height: \(H = \frac{(20 \cdot \sin 30^\circ)^2}{2 \cdot 9.8} \approx 5.10 \text{ m}\)
Range: \(R = \frac{20^2 \cdot \sin 60^\circ}{9.8} \approx 35.3 \text{ m}\)
Practice Problems
- A projectile is fired horizontally at 15 m/s from a 45 m high cliff. Find the time to hit the ground and horizontal distance traveled.
- A stone is thrown at 25 m/s at 45°. Determine the maximum height and range.
- A ball is thrown with 12 m/s at 60°. Find the time it reaches maximum height and the height.
- Given a projectile with initial velocity 10 m/s at 30°, find horizontal and vertical components of velocity after 1 second.