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Maximum range at 45°
Intermediate
On level ground the range is greatest at 45°.
R_max = u²/gSince sin2θ is largest (1) at θ = 45°, the maximum range is u²/g. With drag or a launch height above the target the best angle becomes smaller than 45°.
Throwing at 40 m/s: maximum range ≈ 163 m.
For a launch from a height the optimal angle is less than 45°.
In more detail
With no air resistance, the maximum horizontal range for a given launch speed occurs at a launch angle of 45 degrees, where sin 2θ equals one.
Real-world uses
- Athletes such as shot putters aim for angles near the optimum, though air resistance shifts it.
- Projectile range calculations for simple launchers use 45 degrees as a baseline.
- Long-jump and shot-put coaches aim for near 45 degrees on level ground when the goal is maximum horizontal distance.
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Frequently asked questions
Why 45 degrees?
Because sin 2θ is maximum, equal to one, when 2θ is 90 degrees.
Does air resistance change the best angle?
Yes. With drag the best angle is usually a little below 45 degrees.
Does 45 degrees always give the best distance?
On level ground without air resistance, yes. In sport, drag and the launch height above the ground change the best angle.
Related topics in Kinematics
Maximum height · Horizontal range · Complementary angles range · Horizontal projectile · Projectile on an incline · Trajectory equation