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Projectile on an incline
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Range along an incline depends on both launch and plane angles.
Choose axes along and perpendicular to the plane: gravity then has components g sinα and g cosα along each. Time of flight becomes T = 2u sin(θ − α)/(g cosα), and the range along the plane follows.
A ball thrown perpendicular to an incline of angle α stays in the air 2u/(g cosα).
Don't use horizontal-ground formulas on slopes.
In more detail
A projectile launched up an incline lands at a different height than on level ground. Taking axes along and perpendicular to the slope simplifies the calculation.
Real-world uses
- Ski jumpers and ramp jumpers face projectile motion from sloping launch surfaces.
- Estimating the landing distance for a ball hit down a hillside.
- A ski jump landing slope is an incline, and the range along the slope depends on the take-off angle and the slope angle.
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Frequently asked questions
Which axes should I use?
Axes parallel and perpendicular to the incline usually simplify the algebra.
Why does the range change on a slope?
Because the landing surface is no longer at the launch height.
Why is the range on a slope different from level ground?
The landing surface is not horizontal, so the vertical drop changes the time of flight and the distance along the slope.
Related topics in Kinematics
Complementary angles range · Maximum range at 45° · Horizontal projectile · Trajectory equation · Velocity at any point of projectile · Effect of air resistance