The discussion focuses on the physics of projectile motion on an inclined plane, specifically how the angle of projection (beta) and the incline angle (alpha) affect flight time. The time of flight is derived using two expressions: \( t_{flight} = \frac{2v_i \sin \beta}{g \cos \alpha} \) and \( t_{flight} = \frac{v_i \cos \beta}{g \sin \alpha} \). The goal is to determine conditions under which the projectile strikes the inclined plane perpendicularly, leading to specific values for the angles and initial velocity (vi). Dimensional analysis is suggested as a method to gain further insights into the problem.
PREREQUISITESStudents and educators in physics, engineers working with projectile dynamics, and anyone interested in the mathematical modeling of motion on inclined surfaces.
A projectile is fired up an inclined plane with an initial speed vi and at an angle beta with respect to the incline, which, in
turn, rises at an angle alpha with respect to the horizontal.
(b) Determine the time of flight, tflight from when the projectile is launched to when it strikes the inclined plane.
Express your answer in terms of alpha, beta, vi and g.
(c) Determine a second expression for the time of flight based on the condition that the projectile strikes the inclined
plane with a final velocity that is perpendicular to the plane. Again, express your answer in terms of alpha, beta, vi and
g.
(d) Using your results from parts (b) and (c), find the angle beta and the values for the initial speed, vi , for which the
final velocity is perpendicular to the inclined plane. If your answer surprises you a clever use of dimensional
analysis may provide some insight.