Two cars, both of mass m, collide and stick together. Prior to the collision, one car had been traveling north at speed 2v, while the second was traveling at speed v at an angle phi south of east (as indicated in the figure). After the collision, the two-car system travels at speed v_final at an angle theta east of north.

Find the speed v_final of the joined cars after the collision.

Express your answer in terms of v and phi.

## Answers (1)

Hello,

the component of initial momentum along the X- and Y- axes: Pix,Piy.

the component of final momentum along the X- and Y- axes: Pfx,Pfy.

Pix=mvcos(φ)=Pfx=2mVf*cos(θ).....(1)

Piy=2mv-mvsin(φ)=Pfy=2mVf*sin(θ)..(2)

From equation (1): Vf=[vcos(φ)]/[2cos(θ)]

cos(θ)=vcos(φ)/2Vf

using the identity:sin(θ)=sqrt[1-cos²(θ)]

sin(θ)=sqrt[1-v²cos²(φ)/4V²f]...(*)

From equation (2): Vf=v[2-sin(φ)]/[2sin(θ)]..(**)

From (

) and (*):Vf=v[2-sin(φ)]/[2sqrt[1-v²cos²(φ)/4V²f]]

we quadruple both sides of the equation,and at the end of simple algebra:

Vf=1/2*v*sqrt[5-4sin(φ)] <===ANSWER