Horizontal and vertical motion are independent. Gravity pulls down and changes nothing sideways.
Step 1: Let's Learn
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The horizontal equation
With no air resistance the horizontal speed never changes, so x = (v cos θ)t.
The vertical equation
Gravity acts downward, giving y = (v sin θ)t − 4.9t² in metres, plus any starting height.
Finding the flight time
Set y = 0 and solve. The positive root is when the projectile lands.
Finding the peak
The path is a parabola in t, so the maximum height occurs exactly halfway through the flight.
The best angle
On level ground and ignoring air, 45° gives the greatest range. It splits the speed evenly between going far and staying up.
Two independent motions at once
Horizontal motion is constant velocity; vertical motion is constant acceleration under gravity. The two are entirely independent, and treating them separately is what makes projectile motion tractable.
The standard equations
x = v₀cos(θ)t and y = v₀sin(θ)t − ½gt² + h₀. Each is a one-dimensional motion; the parameter t links them. Resolving the initial velocity into components is the first step every time.
The questions and how to answer them
Maximum height comes from the vertex of the vertical equation. Range comes from setting y to zero and substituting the time into x. Both reduce to solving a quadratic in t.
The model ignores air resistance
Real projectiles fall short of the predicted range, and the discrepancy grows with speed. The parabolic model is an idealisation — accurate for a thrown ball, badly wrong for an artillery shell. Knowing a model's limits is part of using it.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Tomas Find a Flight Time
A ball leaves the ground at 20 m/s at 30° above the horizontal.
- Step 1
The vertical component is 20 sin 30° = 10 m/s.
Step 4: Your Turn
Practice makes it stick.
The Components
Problem 1 of 2
A ball is launched at 20 m/s at 30°. What is the vertical component, in m/s?
m/s
The Halfway Point
Problem 2 of 2
A projectile is in the air for 4 seconds. At what time, in seconds, does it reach its highest point?
seconds
Up and Across
1 of 8
Launch at 40 m/s at 0°. What is the vertical component, in m/s?
2 of 8
Launch at 40 m/s at 90°. What is the horizontal component, in m/s?
3 of 8
Launch at 10 m/s at 30°. What is the vertical component, in m/s?
4 of 8
Flight time 6 seconds. When is the peak, in seconds?
5 of 8
Horizontal speed 15 m/s for 4 seconds. What is the range, in metres?
6 of 8
Which launch angle gives the greatest range on level ground, in degrees?
7 of 8
Sort each quantity by which motion it belongs to.
Tap something to move it.
- Empty
- Empty
8 of 8
Horizontal speed 20 m/s for 3 seconds. Range in metres?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Launch at 30 m/s at 30°. What is the vertical component, in m/s?
Question 2 of 2
What happens to the horizontal speed during flight, ignoring air resistance?
What You Learned
- Horizontal and vertical motion are independent of each other.
- x = (v cos θ)t and y = (v sin θ)t − 4.9t².
- The peak occurs halfway through the flight, and 45° maximises range on level ground.