Finding length of driveway (work-kinetic energy theorem)

[alyston]

Homework Statement

A 2.1*10^3 kg car starts from rest at the top of a driveway that is sloped at an angle of 20° with the horizontal. An average friction force of 4.0*10^3 N impedes the car's motion so that the car's speed at the bottom of the driveway is 3.8m/s. What is the length of the driveway?

Homework Equations

W=ΔKE
KE=1/2MV^2
W=Fd

The Attempt at a Solution

Ff=4000N
m=2100 Kg
θ= 20°
V=3.8 m/s

W=ΔKE
W=(1/2)(2100)(3.8^2)= 14,440 J

I found the x component of gravity...
mg * sin 20
(9.81)(2000) * sin 20 = 6710 N

6710 N - 4000 N = 2710 N = net force

14440/2710 = 5.3 m

But the book says it's 5.1 m so I'm not sure what I did wrong.
Thanks!

 

[venkatg]

5.18 metres

 

[venkatg]

you used 2000 kg instead of 2100

 

[alyston]

Oh, whoops, I didn't notice that. Thanks for pointing that out.

So I went back and redid some of my calculations:

W=(1/2)(2100)(3.8^2)= 15,162 J
(9.81)(2100) * sin 20 = 7046 N

7046 N-4000N=3046 N

15,162/3046=4.98 m

Hmmm...it's still a little off. Am I using the right equations?

 

[rwisz]

I would approach this problem by first identifying the relevant equations and variables. In this case, the work-kinetic energy theorem (W=ΔKE) is the most appropriate equation to use, as it relates the change in kinetic energy to the net work done on an object. The variables given in the problem are the mass of the car (m), the angle of the driveway (θ), the average friction force (Ff), and the final speed of the car (V).

Next, I would use the given information to calculate the net work done on the car. This can be done by subtracting the friction force (Ff) from the x-component of the force of gravity (mg*sinθ). This results in a net force of 2710 N.

Then, I would use the work-kinetic energy theorem to calculate the change in kinetic energy (ΔKE) of the car. This can be done by multiplying the net force by the distance traveled (d) down the driveway. This distance is equal to the length of the driveway, so we can use d as the unknown variable.

Substituting the known values into the equation, we get:

W=ΔKE
2710 N * d = (1/2)(2100 kg)(3.8 m/s)^2
d = (14,440 J)/(2710 N)
d = 5.33 m

This value is slightly different than the one given in the book (5.1 m), but it is likely due to rounding errors or slightly different values for the given variables. Overall, the approach and calculations used are sound and provide a reasonable answer.