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MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Department of Physics

8.01

W07D3-3 Energy Diagram Solution

A particle of mass m moves in one dimension. Its potential energy is given by

U(x) =!U0e

!x2/a

2

,

where U0and aare positive constants. The mechanical energy Eof the particle is

constant such that !U0

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E =U(x = a) =!U0e

!a2/a

2

=!U0e

!1 .

When the particle is at the origin, the kinetic energy is given by

K((x =

0)= E!U

(x =

0)=!U

0e

!1!

(!U

0 )=U

0 (1! e

!1

)=

1

2mv

2

(x =

0) .

Thus the speed of the particle at x = 0 is

v(x =0) =2

mU

0(1! e

!1) .