Projectile Motion: Formulas, Worked Example and Calculator
Projectile motion explained with a calculator: range, maximum height and time of flight formulas, a worked example and why 45 degrees gives the longest range.
Free Fall Calculator and Formulas
Free fall explained with a calculator: time to fall, impact speed and distance fallen, the equations of motion with g = 9.81 m/s², examples and air resistance.
Key takeaways
- In free fall (ignoring air resistance), objects accelerate at g ≈ 9.81 m/s².
- Time to fall from height h: t = √(2h/g). Speed on impact: v = √(2gh).
- Real objects are slowed by air resistance and reach terminal velocity.
On this page
Drop a ball and it speeds up as it falls. Without air resistance, every object near Earth’s surface accelerates downward at the same rate: about 9.81 m/s².
Free fall calculator
Enter a drop height (starting from rest).
Free fall calculator
The equations
Starting from rest:
- Velocity after time t: v = g t
- Distance fallen after time t: d = ½ g t²
- Time to fall height h: t = √(2h ÷ g)
- Speed after falling h: v = √(2 g h)
Worked example
A stone dropped from 20 m: t = √(2 × 20 ÷ 9.81) ≈ 2.02 s; v = √(2 × 9.81 × 20) ≈ 19.8 m/s.
Does mass matter?
Without air resistance, no. A hammer and a feather fall together on the Moon, as Apollo 15 astronaut David Scott famously demonstrated in 1971.
Air resistance and terminal velocity
In air, drag grows with speed until it balances weight, and the object stops accelerating: terminal velocity. A skydiver’s terminal velocity is very different from a raindrop’s.
Related
Free fall is a special case of projectile motion; see our projectile motion calculator. For the energy view, see kinetic vs potential energy, and for the underlying law, Newton’s laws of motion.
Frequently asked questions
What is the acceleration due to gravity?
About 9.81 m/s² near Earth’s surface, often rounded to 9.8 or 10 in exams.
Do heavier objects fall faster?
Not in a vacuum. In air, differences come from air resistance, not mass alone.
What is terminal velocity?
The constant speed reached when air resistance balances the force of gravity.
Sources
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