Can you find the velocity of a motorcycle launching off a parabolic ramp?

[djiberal]

Homework Statement

Let's say that a ramp is modeled using the equation y = cx², where c is a constant. Assuming a motorcycle starts at speed v_i, displacement x_i=0 and launches off the ramp at a horizontal displacement of x_f, what is the motorcyclists speed and direction?

Homework Equations

[/B]
y = cx²
gradient = m = tan(θ)
F = gsin(θ)
F = ma

The Attempt at a Solution

The gradient at each point, x, is given by:
dy/dx = m = 2cx

For an inclined plane, m = tan(θ), so from above:
tan(θ) = 2cx
θ = tan^-1(2cx)

The force down the slope at any point is F = gsin(θ), so:
F = gsin(tan^-1(2cx)) = (2cxg)/√(1+4x²)
a = F/m = (2cxg)/(m√(1+4x²))

V = ∫adt, but the acceleration is in terms of x so I'm not sure what to do... I obviously need to find a way to link the displacement to the velocity, but I'm not sure how.

 

[haruspex]

Hint: You don't need to deal with forces and accelerations.