Ohm’s Law: Calculator and Examples
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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.
Key takeaways
- Split the motion: horizontal velocity stays constant, vertical velocity changes because of gravity.
- On level ground: time of flight T = 2v sinθ ÷ g, maximum height H = v² sin²θ ÷ 2g, range R = v² sin2θ ÷ g.
- Without air resistance, 45° gives the maximum range on level ground.
On this page
A projectile is anything launched into the air and then left to move under gravity alone: a thrown ball, a long jumper, a water jet. The path it follows is a curve called a parabola. The key to solving any projectile problem is to treat horizontal and vertical motion separately.
The core idea
Ignoring air resistance:
- Horizontal motion: no force acts horizontally, so horizontal velocity stays constant.
- Vertical motion: gravity pulls down at g ≈ 9.81 m/s², so vertical velocity changes steadily.
If an object is launched at speed v and angle θ above the horizontal:
- Horizontal component: vₓ = v cosθ
- Vertical component: vᵧ = v sinθ
The formulas (level ground)
| Quantity | Formula |
|---|---|
| Time of flight | T = 2v sinθ ÷ g |
| Maximum height | H = v² sin²θ ÷ 2g |
| Horizontal range | R = v² sin2θ ÷ g |
| Horizontal position at time t | x = v cosθ × t |
| Vertical position at time t | y = v sinθ × t − ½gt² |
Projectile motion calculator
Enter a launch speed and angle. The calculator assumes level ground and no air resistance.
Projectile motion calculator
Worked example
A ball is kicked at 20 m/s at 30° above the horizontal. Take g = 9.81 m/s².
- Components: vₓ = 20 cos30° ≈ 17.32 m/s; vᵧ = 20 sin30° = 10 m/s.
- Time of flight: T = 2 × 10 ÷ 9.81 ≈ 2.04 s
- Maximum height: H = 10² ÷ (2 × 9.81) ≈ 5.10 m
- Range: R = 17.32 × 2.04 ≈ 35.3 m
Check it with the calculator above: enter 20 m/s and 30°.
Why 45° gives the longest range
The range formula contains sin2θ, which reaches its maximum value of 1 when 2θ = 90°, so θ = 45°. Angles equally above and below 45°, such as 30° and 60°, give the same range: the higher launch stays in the air longer but travels more slowly across.
In real life, air resistance and launch height change this. Athletes throwing from shoulder height often get the best distance at angles a little below 45°.
Common mistakes
- Mixing degrees and radians on your calculator.
- Using the full speed instead of its components.
- Forgetting that vertical velocity is zero at the top, not the total velocity: the horizontal component is still there.
- Using level-ground formulas when the landing point is higher or lower than the launch point.
These formulas are Newton’s second law in action; see Newton’s laws of motion for the foundations.
Frequently asked questions
Does mass affect projectile motion?
Without air resistance, no. All objects fall with the same acceleration, so mass doesn’t appear in the formulas.
What is the path of a projectile called?
A parabola, when air resistance is ignored.
How do I solve a horizontal launch problem?
Set θ = 0. The initial vertical velocity is zero, so use the fall height to find the time, then multiply by the horizontal speed to find the distance.
Sources
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