A 10.0-g marble slides to the left with a velocity of magnitude 0.500 m/s on the

Question

A 10.0-g marble slides to the left with a velocity of magnitude 0.500 m/s on the frictionless, horizontal surface of an icy, New York sidewalk and has a head-on, elastic collision with a larger 20.0-g marble sliding to the right with a velocity of magnitude 0.150 m/s. Find the velocity of each marble (magnitude and direction) after the collision. (Since the collision is head-on, all the motion is along a line. Take right as the positive x direction.) ____________ m/s (smaller marble) ____________ m/s (larger marble) Calculate the change in momentum (that is, the momentum after the collision minus the momentum before the collision) for each marble. ____________ kg-m/s (smaller marble) ____________ kg-m/s (larger marble) Calculate the change in kinetic energy (that is, the kinetic energy after the collision minus the kinetic energy before the collision) for each marble. ____________ J (smaller marble) ____________ J (larger marble)

Explanation / Answer

10 0.5 20 0.15

m vi + m vi = m vf + m vf

For an elastic collision, kinetic energy is conserved. That is:

0.5 m vi² + 0.5 m vi² = 0.5 m vf² + 0.5 m vf²

Combining the above equations gives a solution to the final velocities for an elastic collision of two objects:

vf = [(m - m) vi + 2 m vi]/[m + m]

vf = [2 m vi (m - m) vi]/[m + m]

substituting the values
m = 10 g
m = 20 g
vi = -0.5 m/s
vi = 0.15

vf = [(0.01 – 0.02)(-0.5) + 2(0.02)0.15]/[0.01 + 0.02]

vf = 0.0011/0.03 = 0.366 m/s

(the direction of velocity of 10 g slider is reversed towards right)

and

vf = [2(0.01)(-0.5) (0.01 – 0.02)(0.15)]/[0.03]

vf = -0.283 m/s

(the direction of velocity of 20 g slider is reversed towards left)

b) change in momentum[(mv)]:

for the 10 g marble = 0.01 [0.5 - (-0.5)] = 0.01 kg-m/s

for the 20 g marble = 0.02[-0.15 - 0.15] = -0.006 kg-m/s

c)

loss of k.e for 10 g slider = (0.5)(0.01)[(0.5)² - (-0.5)²] = 0 Joules

since this is an elastic collision also equal to k.e gained by the stationary slider = 0 J

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