Speed of each vehicle after the collision?

[M Malone]

Homework Statement

"Before the collision, vehicle #1 was traveling east and vehicle #2 was traveling north.

• The driver of vehicle #1 states he was traveling 35 mph as he approached the intersection. He continues to state that vehicle #2 ran the stop sign, pulling out in front of him and causing him to crash.
• The driver of vehicle #2 states that he stopped at the stop sign before pulling out, and did not see vehicle #1 until the moment of impact. (From the stop sign, vehicle #2 would have traveled 30 feet to the point of impact.

After the collision, both vehicles experience wheel lock due to crash damage and skidded over asphalt (μk = 0.72) followed by grass (μk = 0.35). Neither surface has any significant incline.

• Vehicle #1 skidded on 20 feet of asphalt and 30 feet of grass before coming to rest. The angle of departure for vehicle #1 was 45 degrees north of east.
  The weight of vehicle #1 including occupants was 4300 lbs.
• Vehicle #2 skidded on 25 feet of asphalt and 35 feet of grass before coming to rest. The angle of departure for vehicle #2 was 35 degrees north of east.
  The weight of vehicle #2 including occupants was 3150 lbs.
  An acceleration test concluded that a vehicle such as vehicle #2 would have a maximum acceleration of 2.0 m/s^2 at the time of the accident.

...What was the speed of each vehicle after the collision?"

Homework Equations

(I'm all over the place, but...)

KE = (1/2)(m)(v^2)
W = Fd(cosθ)
W = ΔKE
F(friction) = (μk)(F(normal)) = (μk)(mg)
KE1 + PE1 + W(external) = KE2 + PE2
J = FΔt = mΔv = Δp
p = mv

The Attempt at a Solution

Originally, I used the fact that the work of friction is equal to the change in kinetic energy, so:

W(f) = ΔKE
(μk)(mg)(d)(cosθ) = (1/2)(m)(v^2)
√[2(μk)(g)(d)(cosθ)] = v

But this doesn't account for the change in surfaces, from asphalt to grass. Also, I'm not even sure if it properly accounts for the two-dimensions of the collision.

I'm aware that this is a 2D elastic collision problem, and understand how to split the x- and y-components of the velocity. However, I don't think I'm approaching this properly.

Any help would be greatly appreciated.
Thank you!

 

[gneill]

You're given the distances traveled on each surface type. Imagine that an accident surveyor laid a measuring tape out along the paths that each vehicle took and recorded the straight-line distances of each surface for both vehicles. So no need to invoke trajectory angles in this part of the question.

For each vehicle write an expression for the total work done by friction from impact to stopping on the grass. The total for a given vehicle will be the sum of the energy it lost on asphalt and on grass.