Projectile Motion — Range, Height, Time of Flight
A projectile is any object launched into the air and left to move under gravity alone — no engine, no rocket, no continued pushing. From a thrown ball to an artillery shell to a basketball shot, every such object follows the same elegant mathematics. Master three equations and you can predict exactly where anything will land, how high it will go, and how long it will take.
1. The Physical Idea
The key insight in projectile motion — discovered by Galileo Galilei in the early 1600s — is that horizontal and vertical motion are completely independent of each other. Gravity acts only vertically. There is nothing acting horizontally (ignoring air resistance). So we treat the two directions separately and combine the results at the end.
- Horizontally: No force acts → constant velocity → x = u·cos(θ)·t
- Vertically: Gravity acts at constant g downward → uniform acceleration → y = u·sin(θ)·t − ½gt²
The horizontal and vertical components of projectile motion are completely independent. Gravity does not affect horizontal motion; horizontal velocity does not affect vertical fall.
This is why two balls — one dropped vertically, one launched horizontally from the same height — hit the ground at exactly the same time.
2. Decomposing Initial Velocity
When a projectile is launched at speed u and angle θ above the horizontal, the initial velocity breaks into two components:
The horizontal component u_x remains constant throughout the flight. The vertical component u_y decreases at rate g (9.81 m/s²) due to gravity — reaching zero at maximum height, then becoming negative as the object falls back down.
| Launch Angle | cos(θ) — Horizontal | sin(θ) — Vertical | Effect |
|---|---|---|---|
| 0° | 1.00 (maximum) | 0.00 | No vertical launch — hits ground instantly |
| 30° | 0.866 | 0.500 | Good range, moderate height |
| 45° | 0.707 | 0.707 | Maximum range (equal horizontal & vertical) |
| 60° | 0.500 | 0.866 | Same range as 30°, greater height |
| 90° | 0.00 | 1.00 (maximum) | Straight up — zero range, maximum height |
3. The Three Key Equations
Maximum Height (H)
At maximum height, the vertical velocity is zero. Using v² = u_y² − 2gH:
Time of Flight (T)
Setting vertical displacement back to zero (for landing at the same height as launch):
Horizontal Range (R)
Range = horizontal speed × time of flight. Using 2sin(θ)cos(θ) = sin(2θ):
4. Why 45° Gives Maximum Range
The range formula is R = u²·sin(2θ)/g. The only angle-dependent factor is sin(2θ). Since sin is maximum when its argument is 90°:
sin(2θ) is maximum when 2θ = 90°, i.e. θ = 45°.
At 45°: sin(90°) = 1, giving R_max = u²/g. Any other angle gives sin(2θ) < 1, meaning a shorter range.
Beautiful Symmetry: Angles that add to 90° give identical ranges. A 30° launch gives the same range as a 60° launch (since sin(60°) = sin(120°)). The low-angle shot arrives faster on a flatter trajectory; the high-angle shot arrives slower from above. Same landing spot, completely different paths.
5. Worked Examples
Problem: A ball is launched at 20 m/s at 30° above horizontal. Find maximum height, time of flight, and range. (g = 9.81 m/s²)
Problem: A golf ball lands 200 m away after being launched at 45°. What was the launch speed? (g = 9.81 m/s²)
6. Common Exam Mistakes
Using the full initial speed in vertical equations. Only u·sin(θ) drives vertical motion. Never substitute the full launch speed u into vertical kinematic equations — decompose first.
Applying the range formula when launch and landing heights differ. R = u²sin(2θ)/g is only valid when the projectile lands at the same height it was launched from. If it lands on a cliff, a slope, or below the launch point, you must use the component equations directly.
Assuming maximum height occurs at half the total time always. This is only true when the launch and landing heights are equal. If the projectile lands at a different height, maximum height occurs at T_up = u·sin(θ)/g, which is not necessarily T/2.
Forgetting that at maximum height, vertical velocity = 0, not total velocity = 0. At the peak, v_y = 0 but v_x = u·cos(θ) is still non-zero. The projectile is still moving horizontally — it is not momentarily stopped.
7. Frequently Asked Questions
Conclusion
Projectile motion is one of the most beautiful applications of Galileo’s independence principle. Horizontal and vertical motion are completely separate; treat them that way and the mathematics flows naturally. The three key equations — H = u²sin²θ/2g, T = 2usinθ/g, R = u²sin(2θ)/g — come directly from kinematics applied to each direction independently. Master these and you can analyse every thrown ball, every launched rocket, every kicked football with complete precision.