Take the weight of the child (mg), and break it into forces down the plane, and normal to the plane.
down the plane mgsinTheta
normal to plane mgCosTheta
So friction is mu*mgCosTheta. Sliding down, the friction opposes it, so
mgSinTheta-mu*mg*CosTheta=mass*acceleration
From this (frictionless means mu is zero), you can write two equations. Remember that velocity at the bottom is directly porportional to the square root of acceleration. see below.
Vf^2=Vi^2 + 2ad
Vf = sqrt (2ad)
So your basic equation will be..
mgSinTheta-mu*mg*CosTheta=mass*constant*velocity^2
Now make your two equations, and solve for mu.
To solve for the coefficient of friction (mu), we can use the following steps:
Step 1: Set up the equations
The given equation is:
mg * sin(theta) - mu * mg * cos(theta) = mass * constant * velocity^2
Step 2: Simplify the equation
Let's simplify the equation by dividing both sides by mass:
g * sin(theta) - mu * g * cos(theta) = constant * velocity^2
Step 3: Solve for mu
Now, isolate mu by moving all the other terms to the other side of the equation:
mu * g * cos(theta) = g * sin(theta) - constant * velocity^2
Divide both sides of the equation by g * cos(theta):
mu = (g * sin(theta) - constant * velocity^2) / (g * cos(theta))
Now, you have the equation to solve for the coefficient of friction (mu) using the given variables and constants.