Question

A car traveling north at 10.0 m/s crashes into a car traveling east at 15 m/s at an unexpectedly icy intersection. The cars lock together as they skid on the ice. The two cars have the same mass. What is their combined speed after the collision?

Answer 1

The first thing you should know is the conservation of the linear momentum
Pi = Pf
We have then that before the shock:
Pi = mvi1 + mvi2 = m (10.0) j + m (15.0) i
We have after the shock:
Pf = 2mvf = 2mv (sinx) j + 2mv (cosx) i
Matching both expressions:
m (10.0) j + m (15.0) i = 2mv (sinx) j + 2mv (cosx) i
Rewriting
(10.0) j + (15.0) i = 2v (sinx) j + 2v (cosx) i
We have 2 equations (components j and i) and two unknowns (angle x and v)
2v (senx) = 10.0
2v (cosx) = 15.0
Resolving:
tanx = (10.0 / 15.0)
x = atan (10.0 / 15.0) = 33.69 degrees
Clearing v
2v (senx) = 10.0
v = (10.0 / 2) * (1 / sen (33.69)) = 9.01 m / s
answer
their combined speed after the collision is 9.01 m / s in the direction 33.69 degrees