Projectile on an upward inclined plane — time of flight expression Given: initial speed u, angle of projection α with the horizontal, and an upward plane inclination β with the horizontal. Select the correct formula for the time of flight (until the projectile strikes the plane).

Setup

• Inclined plane makes angle β with the horizontal.
• Projectile launched at speed u and angle α with the horizontal.
• Axes chosen: tangential to plane (‖) and normal to plane (⊥).

ComponentsInitial velocity along the plane: u_∥ = u cos(α − β)Initial velocity normal to the plane: u_⊥ = u sin(α − β)Acceleration along the plane: a_∥ = g sin β (down the plane)Acceleration normal to the plane: a_⊥ = g cos β (toward the plane)

Step-by-Step (normal motion) Normal displacement y_⊥(t) = u sin(α − β) t − &frac{1}{2} g cosβ \, t^2 Impact with the plane occurs when y_⊥(t) returns to zero (excluding t = 0): u sin(α − β) t − &frac{1}{2} g cosβ \, t^2 = 0 \Rightarrow t = \dfrac{2u \sin(α − β)}{g \cosβ}

Common pitfalls

• Using g instead of g cosβ for the normal component.
• Confusing (α − β) with (α + β); the effective launch angle relative to the plane is (α − β).

Final Answer(2u sin(α − β)) / (g cos β)