Projectile motion describes how an object moves after it is launched and gravity is the only significant force acting on it. A thrown ball, a kicked soccer ball, and water from a fountain all follow the same basic physics when air resistance is small. The path is a parabola because horizontal motion stays uniform while vertical motion accelerates downward.
Understanding range and trajectory helps predict where an object will land and how high it will rise.
Key Facts
- Horizontal velocity is constant when air resistance is ignored: vx = v0 cos(theta).
- Vertical velocity changes due to gravity: vy = v0 sin(theta) - gt.
- Projectile position equations are x = v0 cos(theta)t and y = v0 sin(theta)t - 1/2 gt^2.
- For launch and landing at the same height, time of flight is T = 2v0 sin(theta) / g.
- For launch and landing at the same height, range is R = v0^2 sin(2theta) / g.
- For launch and landing at the same height, maximum height is H = v0^2 sin^2(theta) / (2g).
Vocabulary
- Projectile
- A projectile is an object that moves through the air under the influence of gravity after being launched.
- Trajectory
- A trajectory is the curved path followed by a projectile.
- Range
- Range is the horizontal distance a projectile travels before it lands.
- Launch angle
- The launch angle is the angle between the projectile's initial velocity and the horizontal direction.
- Time of flight
- Time of flight is the total time a projectile remains in the air from launch to landing.
Common Mistakes to Avoid
- Using the total launch speed as the horizontal speed is wrong because only the horizontal component stays constant, so use vx = v0 cos(theta).
- Forgetting that vertical acceleration is downward is wrong because gravity makes ay = -g if upward is chosen as positive.
- Applying R = v0^2 sin(2theta) / g when launch and landing heights are different is wrong because that formula assumes equal starting and ending heights.
- Assuming 45 degrees always gives maximum range is wrong because 45 degrees is only the ideal result for level ground with no air resistance.
Practice Questions
- 1 A ball is launched from ground level at 20 m/s at an angle of 30 degrees. Ignoring air resistance and using g = 9.8 m/s^2, find its time of flight and range.
- 2 A projectile is launched from ground level at 15 m/s at an angle of 60 degrees. Using g = 9.8 m/s^2, calculate its maximum height.
- 3 Two projectiles are launched from the same point with the same speed on level ground, one at 30 degrees and one at 60 degrees. Explain why their ranges are the same but their maximum heights are different.