Question

A block is at the base of of a ramp of angle 35°. The ramp is 5 m. if the block is initially given a velocity of 12 m/s up the ramp, how far from the ramp will the block land?

Answer 1

I'm surprised nobody has answered yet.

I'll guide you through the steps with intermediate results to check against, but I'll leave the actual work to you.

You're given the length and angle of the ramp, the initial velocity of the block, and the coefficients of friction.

Note: You don't have the mass, so keep the mass as a variable for now.

The block is slowing down. The forces acting to slow the block are Kinetic friction and gravity parallel to the ramp.

Since the block is moving, you only need the kinetic friction.
Fk = K*Fn
Fn = m*g*cos(35), then plug in to solve for Fk.
Fk = 0.2 * m * g * cos(35)

Solve for g parallel to the ramp:
Fg_para = m * g * sin(35)

add these forces
F = Fk + Fg_para = [0.2 * m * g * cos(35)] + [m * g * sin(35)]
F = m * g * (0.2 * cos(35) + sin(35))

Find the acceleration of the block:
F = mA ==> A = F/m
A = m * g * (0.2 * cos(35) + sin(35)) / m

A = g * (0.2 * cos(35) + sin(35))
(notice that the mass term drops out; you didn't need it after all.)

Interim solution:
If you've done everything correctly so far, you should have A = -7.23 m/s^2

Now solve for the time it takes to get to the top of the ramp.
Remember, X = Xo + VoT + .5 AT^2
Plug in X = 5, Xo = 0, V0 = 12 m/s, and A = -7.23 m/s^2
With a little re-arranging, you should have a simple quadratic. Find the roots.
Note that you will find two roots. You want the smaller of the two. The larger represents what would happen with a longer ramp and the frictional force replaced by a steady non-frictional force.

Once you have the time to the top of the ramp, solve for velocity.
V = Vo + AT

Interim solution:
If you've done everything right so far, you should have V = 8.46 m/s

From this point, you have a block travelling on a ballistic trajectory. This is basic freefall physics.

Solve for initial height.
H = Lramp * sin(35) = 5 sin(35)

Solve for initial velocity components:
Vhoriz = V cos(35)
Vvert = V sin(35)

Determine the time to impact:
Remember, X = Xo + VoT + .5 AT^2
The acceleration is due only to gravity here. Don't forget that it is negative.
Plug in the numbers and solve the resulting quadratic for T. You should have one positive and one negative root. Obviously, you want the positive root.

Now that you have T, plug in the horizontal velocity to solve for distance.
D = Vhoriz * T

If you've done everything right, you should have D = 9.77 meters.

The slight variation from your expected answer of 9.79 meters is likely due to differing levels of precision in the original calculation.