Step 1
Split the initial velocity
The initial velocity v0 splits into horizontal and vertical components. The triangle uses the same angle as the launch activity.
vₓ = v0 cos(θ)vᵧ = v0 sin(θ)
Interactive Lesson
Why 45° Goes Farthest
This short lesson builds the range formula one piece at a time. The angle control in the launch activity also updates the velocity triangle and time in the air.
Step 2
Time comes from vertical motion
The vertical component vᵧ = v0 sin(θ) determines how long gravity takes to bring the ball back down.
time in air: T = 2v0 sin(θ)g
A larger vertical component gives more time in the air.
Step 3
Range multiplies both ideas
Range is horizontal speed multiplied by time. That is where both cosine and sine enter.
R = vₓT
R = v0 cos(θ) · 2v0 sin(θ)g
R(θ) = v02g sin(2θ)
Range: 0.0 m
Max height: 0.0 m
Time: 0.00 s
Try: Launch at least 3 angles, including 30°, 45°, and 60°, to compare their ranges.
Explanation
- Start: Launch at least 3 angles, including 30°, 45°, and 60°, to compare their ranges.
Final range function
R(θ) = v02g sin(2θ)
- v0 is initial velocity, and g is acceleration due to gravity.
- v02 appears because launch speed affects both horizontal speed and time in the air.
- sin(2θ) comes from multiplying cos(θ) by sin(θ).
- The largest range happens at 45° because sin(90°) = 1.