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Aristotle
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#1
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Q1) A small smooth sphere of mass 3kg moving on a smooth horizontal plane with a speed of 8m/s collides directly with a sphere of mass 12kg which is at rest. Given that the spheres move in opposite directions after the collision, obtain the inequality satisfied by e.

Q2) A sphere of mass m is moving with a speed V along a horizontal straight line. It collides with an identical sphere of mass m moving along the same straight line in the same direction with speed u (u < V). Show that the magnitude of the impulse on either of the spheres is:

0.5m(1 + e)(V - u)

where e is the coefficient of restitution between the two spheres.

Answer:

Q1) e > 0.25

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Gaz031
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(Original post by Aristotle)
Q1) A small smooth sphere of mass 3kg moving on a smooth horizontal plane with a speed of 8m/s collides directly with a sphere of mass 12kg which is at rest. Given that the spheres move in opposite directions after the collision, obtain the inequality satisfied by e.
Let speeds after collision be A and B:
cons momentum: (3)(8) = 3A + 12B
8 = A + 4B (1)

e = (B-A)/(8)
8e = B-A (2)

Adding (1) and (2)

8(e+1) = 5B
B = (8/5)(e+1)

From (1) A = 8-4B
A =8 - (32/5)(e+1)

B>0. Hence e>-1.
A<0. 8 - (32/5)(e+1) < 0
8 < (32/5)(e+1)
(40/32) - 1 < e
e > 1/4
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Gaz031
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Q2) A sphere of mass m is moving with a speed V along a horizontal straight line. It collides with an identical sphere of mass m moving along the same straight line in the same direction with speed u (u < V). Show that the magnitude of the impulse on either of the spheres is:

0.5m(1 + e)(V - u)

where e is the coefficient of restitution between the two spheres.
Cons momentum:
mV + mu = mA + mB
V + u = A + B (1)

e = (B-A)(V-u)
e(v-u) = B-A (2)

(1)+(2) gives: V + u + e(v-u) = 2B
2B = V + u + ev - eu
B = 0.5[V + u + ev - eu]

I = m[B-u]
I = m[0.5V + 0.5u + 0.5ev - 0.5eu - u]
I = m[0.5V + 0.5ev - 0.5eu - 0.5u]
I = 0.5m[V + ev - eu - u]
I = 0.5m(1+e)(v-u)
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