A block starts from rest at a height of 3.1 m on a fixed inclined plane. The acceleration of gravity is 9.8 m/s2 .
If the block continues to slide on the ground with the same coefficient of friction, how far
will the block slide on the ground until coming to rest?
µ = .22
θ = 25° (angle of the ramp's incline)
Height of block on ramp = 3.1 meters
mass of block = 6.6kg
ΣF = ma
Position(time) = Position initial + Velocity initial X time + .5(accelaration)(time)
The Attempt at a Solution
The first part of this problem is to determine the speed of the block at the bottom of the ramp. Not going into too much detail as to how I solved that, I determined that the blocks speed at the bottom of the ramp is 5.662 m/s, and that the answer is correct. I also found that the acceleration of the block is 2.187 m/s^2.
The part I cannot figure out is how far the block will slide. I started by saying:
accelartion in angled direction = 2.187 m/s^2
acceleration in horizontal direction = 2.187cos(25) = 1.982 m/s^2
Velocity in angled direction = 5.662 m/s
Velocity in horizontal direction = 5.662cos(25) = 5.132 m/s
I know when the block is on the ground, it has 4 forces: Sliding force of the block, Force of Kinetic Friction, mass*gravity, and normal force. Since there is no incline, normal force is equal to mass*gravity.
Force of kinetic friction = µ*normal force = (.22)(9.8)(6.6) = 14.23 N
acceleration of kinetic friction = µ*gravity = (.22)(9.8) = 2.156
acceleration of the block = acceleration in the horizontal direction = 1.982 m/s^2 (from above). Since acceleration of kinetic friction is opposite the direction of the block's acceleration, I subtract those 2 values: acceleration = 1.982 - 2.156 = -.174 m/s^2.
Using the kinematic equation listed above I see that:
Distance = Initial pos + Initial Velocity*time + .5*acceleration*time^2 = 0 + 5.132*time - .087*time^2
Velocity = 0 = Initial velocity + acceleration*time = 5.132 -.174t => t = 29.5 seconds
Distance = 0 + 5.132(29.5) - .087(870.25) = 151.394 -75.712 = 75.682 meters
I do not think I am doing this correctly. Can anyone point me in the right direction?