Two carts on a straight track collide head on. the first cart was moving at 3.6 m/s in the positive x direction and the second was moving at 2.4 m/s in the opposite direction. after the collision, the second car continues moving in its initial direction of motion at 0.24 m/s. if the mass of the second car is 5.0 times that of the first, what is the mass and final velocity of the first car?

vf₁  = -7.2 m/s :  The first car moves in the negative x direction after the collision.

Explanation:

Theory of collisions  

Linear momentum is a vector magnitude (same direction of the velocity) and its magnitude is calculated like this:  

p=m*v  

where  

p:Linear momentum  

m: mass  

v:velocity  

There are 3 cases of collisions : elastic, inelastic and plastic.  

For the three cases the total linear momentum quantity is conserved:  

P₀ = Pf   Formula (1)  

P₀ :Initial linear momentum quantity  

Pf : Final linear momentum quantity  

Data

m₁ = m kg : mass of the first car

m₂= 5m kg : mass of the second car

v₀₁ = 3.6 m/s : Initial velocity of m₁  , to the +x axis :

v₀₂= -2.4 m/s m/s : Initial velocity of m₂ ,  to the -x axis

vf₂= -0.24 m/s m/s : Final velocity of m₂ ,  to the -x axis

Problem development

We appy the formula (1):

P₀ = Pf  

m₁*v₀₁ + m₂*v₀₂ = m₁*vf₁ + m₂*vf₂  

We assume that the first car moves in the positive x direction after the collision., so, the sign of the final speeds is positive:

(m)*( 3.6) + (5m)*( -2.4) = (m)*vf₁ +(5m)*(- 0.24)

3.6m  - 12m=  m*vf₁ - 1.2m

We divided by m both sides of the equation

3.6 - 12= vf₁ -1.2

3.6 - 12 +1.2 = vf₁

-7.2  = vf₁

vf₁  = -7.2 m/s  : The first car moves in the negative x direction after the collision.