Exercise 8.43 A 10.0- marble slides to the left with a velocity of magnitude 0.4

Question

Exercise 8.43
A 10.0- marble slides to the left with a velocity of magnitude 0.400 on the frictionless, horizontal surface of an icy New York sidewalk and has a head-on, elastic collision with a larger 30.0- marble sliding to the right with a velocity of magnitude 0.200 . Let be to the right. (Since the collision is head-on, all the motion is along a line.)
TURKY AL-HAJRY
Part A
Find the magnitude of the velocity of 30.0- marble after the collision.
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Part B
Find the magnitude of the velocity of 10.0- marble after the collision.
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Part C
Find the direction of the velocity of each marble after the collision.
The 10.0 g marble is moving to the left and the 30.0 g marble is moving to the right.
The 30.0 g marble is moving to the left and the 10.0 g marble is moving to the right.
Both marbles are moving to the right.
Both marbles are moving to the left.





Part D
Calculate the change in momentum (that is, the momentum after the collision minus the momentum before the collision) for 30.0- marble.
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Part E
Calculate the change in momentum for 10.0- marble.
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Part F
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Part G
Calculate the change in kinetic energy (that is, the kinetic energy after the collision minus the kinetic energy before the collision) for 30.0- marble.
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Part H
Calculate the change in kinetic energy for 10.0- marble.
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Part I
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Explanation / Answer

Conservation of linear momentum.
m1v1i + m2v2i = m1v1f + m2v2f
Collision is head on AND elastic; conservation of kinetic energy simplifies into this form:
v2f - v1f = v1i - v2i
They want the 30g one first; let that since it is positive.

Solve for v2f and substitute into equation one.
v2f = v1(i+f) - v2i
Substitute
m1v1i + m2v2i = m1v1f + m2(v1i + v1f - v2i)
v1f(m1+m2) = m1v1i + 2m2v2i - m2v1i
Divide all that by m1+m2
v1f = [m1v1i + 2m2v2i - m2v1i]/(m1+m2)
Solve that equation for the speed. Don't forget to put negative signs where appropriate! Remember, I specificed here that the 30g = m1. (assuming it's right, i'm playing fast and loose with the algebra here. check my work!)

The 10g ball can be solved using conservation of momentum after you solve the above equation.
m1v1i + m2v2i = m1v1f + m2v2f
v2f = (m1v1 + m2v2 - m1v1f)/m2

This equation is asking for dP, or the change in momentums of mass 1. m2v2f - m2v2i ; use this after solving for v2f in the above.

Same thing here. m1v1f - m1v1i.

We have this equation actually, but let's make it formal.
1/2m1v1f^2 - 1/m1v1i^2 . Solve.

same thing here too
1/2m2v2f^2 - 1/2m2v2i^2

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