how would i solve to find the distance and time it takes for a projectile to hit an incline plane?

Question

Hi so I have a question. It relates to projectile motion on an incline ramp. it goes as follows: A projectile is launched from point O at an angle of 22 degrees with an initial velocity of 25m/s up an incline plane that makes an angle of 10 degrees with the horizontal. The projectile hits the incline plane at point M.

a) find the time it takes for the projectile to hit the incline plane

b) Find the distance it travelled from point O-M

Answer 1

The equations of position with time for the projectile:

y = v_y0t -1/2 gt² and x = v_x0t or y in terms of x:

y = (v_y0/v_x0)x - 1/2 (g/v_y0²) x²

The incline is given as y = x(tan(φ)) where φ is the angle of the incline.

So you have a quadratic equation in x:

(g/(2v_x0²)) x² + ((-v_y0/v_x0) + tanφ)x = 0 Note that x = 0 is solution. factor out an x and solve.

Note that the ratio of v's is tanθ₀ and v_x0 = v₀cosθ₀