Projectile Motion: A Visual Explanation of Range and Maximum Height
Understand projectile motion by splitting launch velocity into horizontal and vertical components, then deriving time, height and range.
Quick answer
Ignoring air resistance, projectile motion combines constant horizontal velocity with vertical acceleration due to gravity. Splitting the launch velocity into v₀cosθ horizontally and v₀sinθ vertically lets us calculate the time of flight, maximum height and horizontal range separately.
Key relationships
Begin by splitting the launch velocity
A projectile launched at speed v₀ and angle θ has one initial velocity vector, but resolving it into components makes the motion easier to analyze. The horizontal component is v₀cosθ. The vertical component is v₀sinθ. Both begin at the same instant and use the same time variable.
With no air resistance, no horizontal force acts after launch, so horizontal acceleration is zero. Gravity supplies a constant downward acceleration g, so only the vertical component changes.
Find the highest point
The projectile rises while its vertical velocity is positive. Gravity reduces that vertical velocity until it reaches zero at the peak. Setting vᵧ = v₀sinθ − gt to zero gives the time to maximum height: t = v₀sinθ/g.
Substituting that time into the vertical-position equation gives H = v₀²sin²θ/(2g). The horizontal component is still present at the peak, so the projectile has not stopped; its instantaneous velocity is horizontal there.
Connect flight time to horizontal range
When launch and landing heights are equal, the descending half mirrors the ascending half. The total time of flight is therefore 2v₀sinθ/g. Multiplying by the constant horizontal speed v₀cosθ gives the familiar range formula.
The identity 2sinθcosθ = sin2θ explains why 30° and 60° produce the same range at the same launch speed and height. The 60° path stays in the air longer and rises higher; the 30° path travels horizontally faster.
What to remember
- Horizontal and vertical motion share the same time but otherwise evolve independently.
- The horizontal velocity stays constant when air resistance is ignored.
- At the highest point, vertical velocity is zero—not the total velocity.
- For equal launch and landing heights, complementary angles give the same range.
Frequently asked questions
Is the velocity zero at maximum height?
Only the vertical component is zero. The projectile still has horizontal velocity when air resistance is ignored.
Why is 45 degrees the maximum-range angle?
For equal launch and landing heights with no air resistance, range is proportional to sin(2θ), whose maximum value occurs when 2θ = 90 degrees.
When does the standard range formula not apply?
It needs modification when launch and landing heights differ or when air resistance is significant.
Watch projectile motion drawn from launch to landing
Play the complete lesson free—drawn and narrated from start to finish, with no account needed.
