Projectile Motion

Range, Maximum Height, and Time of Flight

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Projectile motion describes any object moving through the air under the influence of gravity alone, following a curved path called a parabola. From a thrown ball to a launched rocket, the same equations govern the flight. The key insight is that horizontal and vertical motion are independent: gravity pulls the object downward at a constant rate while the horizontal velocity stays constant (ignoring air resistance).

The launch angle determines the shape of the trajectory. At 45 degrees you get maximum range on flat ground. Lower angles produce flatter, shorter flights; higher angles send the projectile higher but bring it back sooner. Understanding these relationships lets you predict exactly where and when a projectile will land.

Key Facts

Horizontal velocity is constant: $v_x = v_0 \cos \theta$
Vertical velocity changes due to gravity: $v_y = v_0 \sin \theta - gt$
Time of flight: $T = \frac{2v_0 \sin \theta}{g}$
Maximum height: $H = \frac{(v_0 \sin \theta)^2}{2g}$
Horizontal range: $R = \frac{v_0^2 \sin(2\theta)}{g}$
Range is the same for complementary angles (30° and 60° give equal range)

Vocabulary

Trajectory: The curved path followed by a projectile through the air.
Launch angle: The angle between the initial velocity vector and the horizontal.
Range: Horizontal distance traveled from launch to landing on the same level.
Apex: The highest point in the projectile's path where vertical velocity equals zero.
Time of flight: Total time the projectile is in the air from launch to landing.

Common Mistakes to Avoid

Treating horizontal and vertical motion as linked - they are independent. Horizontal speed does not change; only vertical speed changes due to gravity.
Using g = 9.8 m/s² as positive when setting up vertical equations. Choose a sign convention and stick to it: if up is positive, g should be negative in your equations.
Forgetting that maximum range on flat ground occurs at exactly 45 degrees only when the launch and landing heights are equal.
Ignoring the two solutions for time when solving quadratic equations - the negative root represents a time before launch and should be discarded.

Practice Questions

1 A ball is kicked at 20 m/s at an angle of 30°. How long is it in the air and how far does it travel horizontally?
2 A projectile reaches a maximum height of 45 m. What was its initial vertical velocity?
3 Two projectiles are launched at 15° and 75° from the same point with the same initial speed. Without calculating, explain why they travel the same horizontal range.

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