A car with engine trouble slides 85 meters up along a 14 degree ramp before coming to a stop. If the coefficient of friction between the tires and the road is 0.9, what was the velocity of the car at the bottom of the ramp?

I'm not looking for a solution which includes Potential Energy. We haven't learned that in class yet, is there a another way to solve this?

The energy way is the easy way but.

Normal force = m g cos 14
force down ramp = m g sin 14 + mu m g cos 14 = m g (mu cos 14+ sin 14)
= m g ( 1.12 )

F = m a
a = F/m = g (1.12) = 10.9 m/s^2 down ramp
say t is time to stop
v = Vi - 10.9 t
0 = Vi - 10.9 t
so
t = Vi/10.9

d = Vi t - (10.9/2) t^2
85 = Vi^2/10.9 - (10.9/2)(Vi/10.9)^2
85 = (Vi^2/10.9 )( 1 - 1/2)
170 = Vi^2/10.9
Vi = 43 m/s

Yes, there is an alternative way to solve this problem without using concepts of potential energy. We can approach this problem using the principles of Newton's laws of motion.

Let's break down the problem step by step and apply Newton's laws to find the velocity of the car at the bottom of the ramp.

1. Identify the forces acting on the car:
- Gravitational force (mg), acting vertically downward
- Normal force (N), which is perpendicular to the surface of the ramp
- Frictional force (f), acting parallel to the surface of the ramp in the direction opposite to the motion of the car
- The force that accelerated the car up the ramp (F), which is unknown

2. Resolve the gravitational force:
The gravitational force acting on the car can be resolved into two components: one perpendicular to the ramp (mgcosθ) and one parallel to the ramp (mgsinθ).

3. Determine the frictional force:
The frictional force can be calculated using the coefficient of friction (μ) and the normal force (N). The equation for frictional force is f = μN.

4. Apply Newton's second law (F = ma):
The net force acting on the car is the difference between the force along the ramp (mgsinθ) and the frictional force (f), which is in the opposite direction:
F_net = mgsinθ - f

We can rewrite this equation as:
F_net = ma

5. Solve for the acceleration (a):
Rearranging the equation from step 4, we can solve for the acceleration:
a = F_net / m

6. Use kinematic equations to find the final velocity:
Now, using the acceleration found in step 5, and the information given in the problem (initial velocity is zero), we can use the appropriate kinematic equation to find the final velocity (v):
v^2 = u^2 + 2as

Since the initial velocity (u) is zero, the equation simplifies to:
v^2 = 2as

Rearranging the equation, we get:
v = √(2as)

7. Substitute the known values and calculate the final velocity:
Plug in the values of acceleration (a), distance (s), and slope angle (θ) to find the final velocity (v).

By following these steps, you should be able to find the velocity of the car at the bottom of the ramp without using the concept of potential energy.