Projectile Motion Calculator: Range, Max Height and Time of Flight
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Enter the launch speed, angle and height.
Launched at 20.00 m/s and 45.00° from a height of 0.00 m on Earth, the projectile flies for 2.88 s, rises to 10.20 m and lands 40.79 m away at 20.00 m/s.
Show the steps
- Split the speed: vx = v cos θ = 14.14 m/s, vy = v sin θ = 14.14 m/s.
- Time to the peak = vy ÷ g = 1.44 s; maximum height = h0 + vy² ÷ (2g) = 10.20 m.
- Time of flight solves h0 + vy t − ½ g t² = 0, giving T = 2.88 s. Range = vx × T = 40.79 m.
Projectile motion
Horizontal distance: x = v cos θ × t
Height: y = h0 + v sin θ × t − ½ g t²
Range (flat ground) = v² sin 2θ ÷ g
- Air resistance is ignored. Real objects travel a shorter distance, especially at high speed.
Estimates for planning and learning, not financial, tax, legal or medical advice. Check figures with your lender, tax preparer or doctor before you act.
About this calculator
A projectile is anything thrown or launched and then moving under gravity alone: a ball, an arrow, a water jet. Its path is a parabola. This calculator finds the range, maximum height, time of flight and impact speed from the launch speed, angle and height.
Choose your speed unit, try different gravities such as the Moon or Mars, and see the path drawn on a graph. Air resistance is ignored.
How to use it
- Enter the launch speed and choose m/s, km/h, mph or ft/s.
- Enter the launch angle above the horizontal, and the starting height in meters.
- Choose the gravity, then read the range, height, time and path.
Projectile motion formulas
- Horizontal speed: vx = v cos θ
- Vertical speed: vy = v sin θ
- Time to the peak: vy ÷ g
- Maximum height: h0 + vy² ÷ (2g)
- Range on flat ground: v² sin 2θ ÷ g
Example
A ball launched at 20 m/s and 45° from the ground has vx = vy = 14.14 m/s, flies for 2.88 s, reaches 10.2 m, and lands 40.8 m away.
Best angle
From level ground, 45° gives the longest range. From a height, the best angle is a little lower.
Sources
Quick check: test yourself
On level ground, which launch angle gives the longest range (no air resistance)?
Correct answer: 45°. Range is v² sin 2θ ÷ g, which is largest when 2θ = 90°.
At the top of its path, a projectile’s vertical velocity is
Correct answer: zero. The projectile stops rising and begins to fall, so vy = 0 at the peak.
On the Moon, gravity is weaker. A ball thrown the same way travels
Correct answer: farther. Smaller g means a longer flight time and a greater range.
Pick an answer for each question.
Common questions
What angle gives the maximum range?
45 degrees from level ground when air resistance is ignored.
Does the calculator include air resistance?
No. It models gravity only, so real objects travel shorter distances.
What is the launch height?
How far above the ground the projectile starts. Use 0 for the ground.
How do I use mph?
Choose mph in the speed unit. The calculator converts it to meters per second.
Why does the Moon give longer range?
Gravity is about one sixth of Earth’s, so the object stays up longer.
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