Table of Contents
Projectile Motion Basics
Projectile motion describes the trajectory of an object launched into the air under the influence of gravity alone (neglecting air resistance). The motion can be decomposed into independent horizontal (constant velocity) and vertical (constant acceleration) components. The maximum height is reached when the vertical component of velocity becomes zero.
The maximum height depends on the initial speed and launch angle. At 90 degrees (straight up), all velocity goes into height. At 0 degrees (horizontal), no height is gained. The maximum range (horizontal distance) occurs at 45 degrees, but the maximum height increases with angle up to 90 degrees. This analysis is fundamental in ballistics, sports science, and orbital mechanics.
Maximum Height Formula
Where Hmax is the maximum height, h0 is the initial height, v0 is the launch speed, θ is the launch angle, and g is gravitational acceleration.
Height at Various Angles (v0 = 50 m/s)
| Angle | Max Height (m) | Range (m) | Flight Time (s) |
|---|---|---|---|
| 15° | 8.5 | 127.4 | 2.6 |
| 30° | 31.9 | 220.7 | 5.1 |
| 45° | 63.7 | 254.8 | 7.2 |
| 60° | 95.6 | 220.7 | 8.8 |
| 75° | 119.3 | 127.4 | 9.9 |
Frequently Asked Questions
Why is the maximum height independent of horizontal velocity?
In projectile motion (without air resistance), the horizontal and vertical components of motion are independent. The horizontal velocity does not affect the vertical motion. Maximum height depends only on the vertical component v0 sin(θ) and gravity. The horizontal component only affects how far the projectile travels during its flight.
How does air resistance affect maximum height?
Air resistance (drag) reduces the maximum height because it opposes the upward motion, slowing the projectile faster. The effect is more significant for light, fast objects with large surface areas (like badminton shuttlecocks) and less significant for heavy, dense objects (like cannonballs). For a baseball thrown at 40 m/s, air resistance reduces the max height by about 15-25%.
What angle gives maximum height?
For maximum height, launch straight up (90 degrees): H = v0^2/(2g). At this angle, all kinetic energy converts to potential energy. However, the range is zero. The optimal angle for maximum range is 45 degrees (without air resistance). With air resistance, the optimal range angle decreases to about 35-40 degrees depending on the object.