Finding initial velocity of car 1 before hitting stationary car 2

[Ellen W.]

Homework Statement

A car hits another car at rest. Debris is found 6.75m from the cars, the two cars stuck together over a distance of 6.75m, there are skid marks a distance of 312m before the car crash, car 1=2946kg, car2=1232kg, coefficient of friction is .72. Find the initial velocity of car 1.

Homework Equations

W=ΔKE=KEf-KEi=fd
m1v1=(m1+m2)v2

The Attempt at a Solution

I assumed the distance after the collision is 6.75m, and the distance before it is 312m.

First I tried to solve for the speed when the cars hit using the work-energy theorem
W=ΔKE=KEf-KEi, KEf=0
KEi=-1/2mv²
W=fd, W=Ff(d), W=(mg)(coefficient of static friction)(d)=-1/2mv²
(2946+1232)(.72)(6.75)(9.81)=1/2(2946+1232)(v²), v=9.76m/s

Then I used that answer in the conservation of momentum equation
m1v1=(m1+m2)v2
2946v1=4178(9.76)
v1=13.85ms=30.98mph

I'm not completely sure what I was supposed to use for my force, since the car was already in motion, but no forces were acting on the car besides friction.I feel like I messed up the part in the work-energy theorem. I ended up not using the stopping distance for car 1 and I think my answer should be greater, but I'm not sure.

 

[Benoit]

I think you're good, but the problem isn't solve yet ! You found the speed of the first car when it hits the other car. Why do you think there are skid marks ?

 

[Kaura]

Friction acts as a decelerating force so by solving for normal force and then the deceleration due to friction you should be able to plug the values of acceleration and distance into the kinematic equation V² = V₀²+2ax and solve for the car's velocity at the moment it began to decelerate and create the skid marks

 

[haruspex]

One small point, you said static friction instead of kinetic. Oh, and you missed a g at one point in typing in your working.

 

[Ellen W.]

Oh, so the friction being solved for is kinetic, since the car tires are skidding as opposed to rolling?

I tried doing what Kaura said, but got a completely different answer then before.
-Ff=ma
Ff=mg(coeff. friction)
-Mgcoff.fricti.=ma
a=-7.0632

9.76²=vo²+2(-7.0632)(312)
Vo²=67.102m/s=150mph
This number seems just as off as before...I'm not sure where to go from here. Could the velocity when hit be wrong?

 

[haruspex]

Oh, so the friction being solved for is kinetic, since the car tires are skidding as opposed to rolling?

I tried doing what Kaura said, but got a completely different answer then before.
-Ff=ma
Ff=mg(coeff. friction)
-Mgcoff.fricti.=ma
a=-7.0632

9.76²=vo²+2(-7.0632)(312)
Vo²=67.102m/s=150mph
This number seems just as off as before...I'm not sure where to go from here. Could the velocity when hit be wrong?

You've reused 9.76, the velocity after collision, instead of using the velocity just before collision.

 

[Ellen W.]

How would I get velocity just before the collision? I wouldn't know either velocity to plug into the equation, nor do I know the time so I can't use the other kinematic equations.

 

[haruspex]

How would I get velocity just before the collision?

You already did that: v1 in your original post.

 

[Ellen W.]

Oh, wow. So I would just plug in 13.85 as initial velocity instead of 9.76? Thank you so much! :)