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.

the average speed is 24.57 km/hr

when two balls of the different masses are dropped from equal height, they reach the ground at the same time .this is because they experience the same acceleration.

the acceleration is independent of mass.

thus when they are dropped they will always maintain the same velocity and travel the same distance (neglect air resistance of course) at the same time.