Engineering & Physics CalcyZone Calculator

Projectile Motion Calculator

Find range, maximum height and flight time of a projectile launched at an angle, with a trajectory table.

Range

40.7886 m

Maximum height

10.1972 m

Time of flight

2.8842 s

Step-by-step working

vₓ = v₀cosθ = 14.1421 m/s, v_y = v₀sinθ = 14.1421 m/s
t = (v_y + √(v_y² + 2gh₀)) / g = 2.8842 s
Range = vₓ t = 40.7886 m
H = h₀ + v_y²/(2g) = 10.1972 m

Trajectory points (no air resistance)
t (s)	x (m)	y (m)
0	0	0
0.288	4.079	3.671
0.577	8.158	6.526
0.865	12.237	8.566
1.154	16.315	9.789
1.442	20.394	10.197
1.731	24.473	9.789
2.019	28.552	8.566
2.307	32.631	6.526
2.596	36.71	3.671
2.884	40.789	0

Mathematical Formula

vₓ = v₀cosθ, v_y = v₀sinθ; t = (v_y + √(v_y² + 2gh₀))/g; R = vₓt; H = h₀ + v_y²/2g (no air resistance).

Overview & Explanation

Without drag, horizontal velocity is constant and vertical motion has constant acceleration g.

How It Works

Split the launch velocity into components.
Solve the vertical motion for landing time.
Range = horizontal speed × time.

Practical Applications

Sports (shot put, football).
Physics labs.

Example 1: 20 m/s at 45° from ground

Maximum range angle.

Inputs

Launch speed (m/s): 20 · Launch angle (°): 45 · Launch height (m): 0

= entered

Step 1

vₓ = v₀cosθ = 14.1421 m/s, v_y = v₀sinθ = 14.1421 m/s

Step 2

t = (v_y + √(v_y² + 2gh₀)) / g = 2.8842 s

Step 3

Range = vₓ t = 40.7886 m

Step 4

H = h₀ + v_y²/(2g) = 10.1972 m

Range

Result

= 40.7886 m

Example 2: 15 m/s at 30° from 2 m

Raised launch.

Inputs

Launch speed (m/s): 15 · Launch angle (°): 30 · Launch height (m): 2

= entered

Step 1

vₓ = v₀cosθ = 12.9904 m/s, v_y = v₀sinθ = 7.5 m/s

Step 2

t = (v_y + √(v_y² + 2gh₀)) / g = 1.7612 s

Step 3

Range = vₓ t = 22.8783 m

Step 4

H = h₀ + v_y²/(2g) = 4.868 m

Range

Result

= 22.8783 m

How it's calculated & sources

vₓ = v₀cosθ, v_y = v₀sinθ; t = (v_y + √(v_y² + 2gh₀))/g; R = vₓt; H = h₀ + v_y²/2g (no air resistance).

HyperPhysics (Georgia State Univ.) – Trajectories

Frequently Asked Questions

Why is 45° best?

From ground level without drag, sin 2θ is maximum at 45°. From a height the optimum is slightly lower.

Air resistance?

Ignored; real ranges are shorter.

Units?

SI: m/s, m, s.

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