Projectile on an InclineJEE Advanced

Range and flight time along a slope — interactive Physics simulation for IIT-JEE.

Concept

Launching up a slope, it pays to rotate your axes onto the incline: gravity then has components both perpendicular (−g cos β, controls flight time) and along the slope (−g sin β, decelerates the range). Maximum range occurs when launch direction bisects the angle between the incline and the vertical.

Key formula

T = \frac{2u\sin(\alpha-\beta)}{g\cos\beta}, \qquad R = \frac{2u^2\sin(\alpha-\beta)\cos\alpha}{g\cos^2\beta}, \qquad R_{max} = \frac{u^2}{g(1+\sin\beta)}

Derivation

In incline axes, initial velocity components are $u\cos(\alpha-\beta)$ (along) and $u\sin(\alpha-\beta)$ (perpendicular). Perpendicular motion: $0 = u\sin(\alpha-\beta)T - \tfrac12 g\cos\beta\,T^2$ gives T.

Substituting T into the along-incline displacement (with the $-g\sin\beta$ deceleration) and simplifying with product-to-sum identities gives R and the maximum at $\alpha = 45° + \beta/2$ .

Scenarios to explore

Projectile on an Incline — Range along a slope — best angle bisects incline & vertical.

Real-world applications

Ski-jump and motocross landing-slope design.
Artillery corrections on sloped terrain.
Drainage of sprinkler jets on embankments.

JEE exam tips

Down-the-incline launch: replace β → −β (range grows, R_max = u²/g(1−sinβ)).
45° + β/2 bisects incline↔vertical — remember geometrically, not by rote.
Perpendicular distance is maximised at T/2, exactly like h_max on flat ground.

Common mistakes

Using the flat-ground formulas T = 2u sinα/g and R = u² sin2α/g on a slope.
Measuring α from the incline in one formula and from the horizontal in another — pick one convention.
Forgetting the along-incline component of gravity when computing range.

Exam traps to avoid

The projectile lands perpendicular to the incline only if v_along = 0 at landing — a separate condition, not automatic.
R is along the slope; the horizontal footprint is R cos β.