A body of mass 2.0 kg makes an elastic collision with another body at rest and continues to move in the original direction but with one-fourth of its original speed. (a) What is the mass of the other body? (b) What is the speed of the two-body center of mass if the initial speed of the 2.0 kgbody was 4.0 m/s?

Short Answer

Expert verified
  1. Mass of another body is,m2=1.2kg
  2. The speed of the two-body center of mass is,Vcom=2.5m/s.

Step by step solution

01

Step 1: Given Data

Mass of first body 1 is,m1=2.0kg

Final speed of first body is,V1f=V1i4

Initial velocity of body 1 is,V1i=4.0m/s

Initial velocity of body 2 is,V1i=0m/s

02

Determining the concept

In an elastic collision, the total energy and linear momentum must be conserved. From the law of conservation of linear momentum, find the mass of another object.According to conservation of linear momentum, momentum that characterizes motion never changes in an isolated collection of objects.

Formulae are as follow:

For conservation of linear momentum,m1V1i+m2V2i=m1V1f+m2V2f

Conservation of kinetic energy,12m1V1i2=12m1V1f2+12m2V2f2

Speed in center of mass is,Vcom=m1V1i+m2V2im1+m2

For elastic collision,V1f=m1-m2m1+m2V1i

Where, m1, m2 are masses and V is velocity.

03

(a) Determining the mass of the body

Let,andare masses of two bodies respectively, where bodyis at rest. Using the law of conservation of momentum,

m1V1i=m1V1f+m2V2fV

Similarly, the total kinetic energy is conserved,

12m1V1i2=12m1V1f2+12m2V2f2

Solving equationis given by equation 9-67,

V1f=m1-m2m1+m2V1i

Solving for,

m2=V1i-V1fV1i+V1fm1

The given condition isV=V1i4, find the second mass to be,

m2=V1i-V1i4V1i+V1i4m1m2=35m1m2=35×2.0m2=1.2kg

Hence,mass of another body is,m2=1.2kg

04

(b) Determining the speed of the two-body center of mass if the initial speed of the 2.0 kg body was

The speed of the center of mass is,

Vcom=m1V1i+m2V2im1+m2Vcom=2.0×4+1.2×02.0+1.2Vcom=8.03.2Vcom=2.5m/s

Hence,the speedof the two-body center of mass is,Vcom=2.5m/s

Therefore, by using the concept of conservation linear momentum, the mass of another body and speed of center of mass can be found.

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