Physics for Scientists and Engineers
Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
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Chapter 8, Problem 64P
To determine

The proof that final velocities are v1f=m1m2m1+m2v1i+2m2m1+m2v2i and v2f=2m1m1+m2v1i+m2m1m1+m2v2i .

Expert Solution & Answer
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Answer to Problem 64P

It is proved that final velocities are v1f=m1m2m1+m2v1i+2m2m1+m2v2i and v2f=2m1m1+m2v1i+m2m1m1+m2v2i .

Explanation of Solution

Formula used:

The expression for the conservation of momentum given by,

  m1v1i+m2v2i=m1v1f+m2v2f

The expression for the conservation of kinetic energy is given by,

  12m1v1i2+12m2v2i2=12m1v1f2+12m2v2f2

Calculation:

The expression for the conservation of momentum is calculated as,

  m1v1i+m2v2i=m1v1f+m2v2fm1(v 1iv 1f)=m2(v 2fv 2i) …… (1)

The expression for the conservation of kinetic energy is calculated as,

  12m1v1i2+12m2v2i2=12m1v1f2+12m2v2f2m1(v 1i2v 1f2)=m2(v 2f2v 2i2)m1(v 1iv 1f)(v 1i+v 1f)=m2(v 2fv 2i)(v 2f+v 2i)

This implies,

  m1( v 1i v 1f )( v 1i + v 1f )m1( v 1i v 1f )=m2( v 2f v 2i )( v 2f + v 2i )m2( v 2f v 2i )(v 1i+v 1f)=(v 2f+v 2i)v1iv2i=v2fv1fv2fv1f=(v 2iv 1i) …… (2)

Further simplify above,

  m2(v 2fv 1f)=(m2)(v 2iv 1i)m2v2fm2v1f=m2v2i+m2v1i …… (3)

Add equation (1) and (3).

  m1v1i+m2v2i+m2v2fm2v1f=m1v1f+m2v2f+(m2v 2i+m2v 1i)(m1m2)v1i+2m2v2i=(m1+m2)v1fv1f=m1m2m1+m2v1i+2m2m1+m2v2i

Multiply equation (2) by m1 .

  m1(v 2fv 1f)=m1(v 2iv 1i)m1vm1v1f=m1v2i+m1+v1i …… (4)

Add equation (1) and (4).

  m1v1i+m2v2im1v2f+m1v1f=m1v1f+m2v2f+m1v2im1v1i(m1m2)v1i+2m2v2i=(m1+m2)v1f2m1v1i+(m2m1)v2i=(m1+m2)v2fv2f=2m1m1+m2v1i+m2m1m1+m2v2i

Conclusion:

Therefore, It is proved that final velocities are v1f=m1m2m1+m2v1i+2m2m1+m2v2i and v2f=2m1m1+m2v1i+m2m1m1+m2v2i .

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Chapter 8 Solutions

Physics for Scientists and Engineers

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