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.