help please?

Question

A 2.1 X 10^3 kg car starts from rest at the top of a driveway that is sloped at an angle of 20.0 degrees with the horizontal. An average friction force of 4.0 X 10^3 N impedes the car's motion so that the car's speed at the bottom of the driveway is 3.8 m/s. What is the length of the driveway?

I got 3.7 but the answer is 5.1 if someone could post how you do this problem i would really appreciate it.

Answer 1

First draw a free body diagram of the car. There are three forces acting on the car:

1) The force due to gravity (this acts straight down), call this Fg
2) The normal force (acts perpendicular to slope), call this Fn
3) The friction force (acts along slope in opposite direction of motion), call this Ff.

The two important forces that determine the acceleration (and ultimately our answer) are Fg and Ff. However, only part of Fg acts in the direction of the cars motion, so only that part contributes to the car's acceleration.

This component of Fg is equal to the magnitude of Fg (mass car * 9.81) times sin(20):

Fgx = Fg*sin(20) = m*g*sin(20) = 7046N

Now, find the net force in the direction of motion:

F = Fgx - Ff = 7046N - 4000N(given in description) = 3046N

The acceleration of the car is the net force on the car divided by the mass of the car:

a = 3046N / 2100kg = 1.5 m/s^2

Now use the equation vf = vi + a*t and solve for t to find out how long the car took to go down the driveway:

t = (vf - vi) / a = (3.8 - 0) / 1.5 = 2.6 s

From here, just plug everything you know into the formula to find how far something travels in a given amount of time with a given acceleration:

d = 0.5*a*t^2 + vi*t = 0.5*1.5*2.6^2 = 5.07m ~ 5.1m

I hope this helps, good luck with the rest of your physics!