Projectile motion calculator
Without air resistance the trajectory is a parabola and 45 degrees gives the greatest range.
m/s2
Horizontal range
40.788 649m
- Maximum height
- 10.197 162
- Time of flight
- 2.884 19 s
- Time to the apex
- 1.442 10 s
- Horizontal velocity
- 14.142 136
- Initial vertical velocity
- 14.142 136
- Impact speed
- 20
Show the working
- Launch speed = 20
- Launch angle = 45
- Launch height = 0
- Gravity = 9.806 65
- theta = rad(45) = 0.785 398
- Horizontal velocity = 20 * cos(0.785 398) = 14.1421
- Initial vertical velocity = 20 * sin(0.785 398) = 14.1421
- Maximum height = 0 + 14.1421^2 / (2*9.806 65) = 10.1972
- Time to the apex = 14.1421 / 9.806 65 = 1.4421
- Time of flight = (14.1421 + sqrt(14.1421^2 + 2*9.806 65*0)) / 9.806 65 = 2.884 19
- Horizontal range = 14.1421 * 2.884 19 = 40.7886
- Impact speed = sqrt(14.1421^2 + (14.1421 - 9.806 65*2.884 19)^2) = 20
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- range = vx x flight time
- apex = h0 + (v sin theta)^2 / 2g
- flight time = (v sin theta + sqrt((v sin theta)^2 + 2 g h0)) / g
Drag is ignored. For a real ball or bullet the optimum angle drops well below 45 degrees.
Launching from above ground level pushes the optimum angle lower still.