A measure of inelasticity in a head-on collision of two objects is the coefficie

Question

A measure of inelasticity in a head-on collision of two objects is the coefficient of restitution, e, defined as e = upsilon'_A - upsilon'_B/upsilon_B - upsilon_A where upsilon'_A - upsilon'_B is the relative velocity of the two objects after the collision and upsilon_B - upsilon_A is their relative velocity before it. (a) Show that e = 1 for a perfectly elastic collision, and e = 0 for a completely inelastic collision. (b) A simple method for measuring the coefficient of restitution for an object colliding with a very hard surface like steel is to drop the object onto a heavy steel plate, as shown in Fig. 7-36. Determine a formula for e in terms of the original height h' and the maximum height h' reached after one collision. Measurement of the coefficient of restitution.

Explanation / Answer

a)

VA = velocity of object A before collision

V'A = velocity of object A after collision

VB = velocity of object B before collision

V'B = velocity of object B after collision

mA = mB = m = mass of each object A and B

using conservation of momentum

mA VA + mB VB = mA V'A + mB V'B

m VA + m VB = m V'A + m V'B

VA + VB = V'A + V'B                      

VA - V'A = - (VB - V'B)                  eq-1

using conservation of kinetic energy

mA V2A + mB V2B = mA V'2A + mB V'2B

V2A + V2B = V'2A + V'2B

V2A - V'2A = - (V2B - V'2B)

(VA - V'A ) (VA + V'A ) = - (VB - V'B) (VB + V'B)

using eq-1

(VA + V'A ) = (VB + V'B)

V'A - V'B = VB - VA

(V'A - V'B)/(VB - VA) = 1

e = 1

for inelastic collision

V'A = V'B= V                      since the two objects move together after collision at same velocity

hence

e = (V'A - V'B)/(VB - VA) = (V - V)/(VB - VA)

e = 0

b)

while the object A falls :

using conservation of energy

kinetic energy just before hitting the surface = initial potential energy at Top

(0.5) m VA2 = m g h

VA = sqrt(2gh)

similarly , for object A after collision

V'A = sqrt(2gh')

also ,

in this case , object B is the steel surface which is remains at rest all the time

hence VB = VB' = 0

e = (V'A - VB') /(VB - VA)

e = (sqrt(2gh') - 0)/(0 - (-sqrt(2gh)))

e = sqrt(h'/h)

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