Results
Range-
Maximum height-
Time of flight-
Live values
Time (t)0.00 s
Horizontal distance (x)0.00 m
Height (y)0.00 m
Horizontal velocity (vx)0.00 m/s
Vertical velocity (vy)0.00 m/s
Speed0.00 m/s
Theory
Projectile motion is the motion of an object that is thrown at some speed and angle and then moves only under gravity. Its path is a parabola.
The initial velocity is split into two parts:
vx = v cosθ (horizontal, stays constant)
vy = v sinθ − g t (vertical, changes with time)
vy = v sinθ − g t (vertical, changes with time)
Position at any time t (origin is on the ground, directly below the launch point):
x = v cosθ · t
y = h + v sinθ · t − ½ g t²
y = h + v sinθ · t − ½ g t²
When the starting height h = 0:
Time of flight: T = 2 v sinθ / g
Maximum height: H = v² sin²θ / (2g)
Range: R = v² sin2θ / g
Maximum height: H = v² sin²θ / (2g)
Range: R = v² sin2θ / g
Key points:
- The range is largest at θ = 45° (when h = 0).
- Angles θ and (90° − θ) give the same range.
- At the highest point vy = 0, but vx stays the same.
- Air resistance is ignored.
Practice questions
Q1. v = 20 m/s, θ = 30°, g = 10 m/s². Find the time of flight, maximum height and range.
T = 2 × 20 × sin30° / 10 = 2 s
H = 20² × sin²30° / (2 × 10) = 5 m
R = 20² × sin60° / 10 = 34.64 m
Q2. An object is thrown at 10 m/s at 45° (g = 10 m/s²). Find the range.
R = v² sin(2 × 45°) / g = 100 × 1 / 10 = 10 m
Q3. A projectile is thrown at 30°. At what other angle will the same speed give the same range?
The range is the same for θ and (90° − θ), so the other angle is 60°.
Tip: Put these values in the sliders above and check your answers in the simulation.