Collisions and conservation of momentum

[sstangle73]

A cart with mass 370 g moving on a frictionless linear air track at an initial speed of 1.7 m/s undergoes an elastic collision with an initially stationary cart of unknown mass. After the collision, the first cart continues in its original direction at 0.42 m/s.
(a) What is the mass of the second cart?
223.3 g
(b) What is its speed after impact?
2.12 m/s
(c) What is the speed of the two-cart center of mass?
?? m/s

Cant get part c?
Any help is appreciated

 

[ehild]

How is the center of mass defined?

ehild

 

[sstangle73]

That is all the information given

 

[ehild]

You have to know the definitions of terms in a problem. What have you learned about center of mass?

Read this: http://online.physics.uiuc.edu/courses/phys211/spring10/Text/ch10.pdf

ehild

 

[rwisz]

.

To calculate the speed of the center of mass, we need to use the conservation of momentum principle. According to this principle, the total momentum of a system remains constant before and after a collision. In this case, the total momentum of the system is the sum of the momentums of the two carts, which can be represented as:

m1v1 + m2v2 = m1v1' + m2v2'

Where:
m1 = mass of the first cart
v1 = initial velocity of the first cart
m2 = mass of the second cart
v2 = initial velocity of the second cart
v1' = final velocity of the first cart
v2' = final velocity of the second cart

Since we know the mass and initial and final velocities of the first cart, we can rearrange the equation to solve for the mass and final velocity of the second cart:

m2v2 = m1v1 + m1v1' - m2v2'
m2 = (m1v1 + m1v1' - m2v2') / v2

Plugging in the values, we get:

m2 = (0.370 kg x 1.7 m/s + 0.370 kg x 0.42 m/s - m2 x 0 m/s) / 2.12 m/s
m2 = (0.629 kg - m2 x 0 m/s) / 2.12 m/s
m2 = 0.629 kg / 2.12 m/s
m2 = 0.296 kg = 296 g

Therefore, the mass of the second cart is 296 g and its final velocity is 2.12 m/s.

To calculate the speed of the center of mass, we can use the formula:

vc = (m1v1 + m2v2) / (m1 + m2)

Plugging in the values, we get:

vc = (0.370 kg x 1.7 m/s + 0.296 kg x 2.12 m/s) / (0.370 kg + 0.296 kg)
vc = (0.629 kg x 1.7 m/s + 0.629 kg x 2.12 m/s) / 0.666 kg
vc = (1.069 kgm/s + 1.333 kg