A particle (mass = m ) moves in one dimension ( x ? 0), under the influence of a

Question

A particle (mass = m) moves in one dimension (x ? 0), under the influence of a conservative force; the potential energy is

U(x) = k(a2?x2)/2 for x ? a

U(x)=0 forx>a
The initial values are x(0) = 0 and v(0)=?{k/m} a.
(A) Calculate the velocity va when m passes the point x = a. Compare va and v(0). (B) Calculate the time ta when m passes the point x = a. Compare ta and a/v(0).

A particle (mass = m) moves in one dimension (x ? 0), under the influence of a conservative force; the potential energy is U(x) = k(a2?x2)/2 for x ? a U(x)=0 forx>a The initial values are x(0) = 0 and v(0)=?{k/m} a. Calculate the velocity va when m passes the point x = a. Compare va and v(0). Calculate the time ta when m passes the point x = a. Compare ta and a/v(0).

Explanation / Answer

a) Ei = Ef

1/2 mv^2 + U(0) = 1/2 mv^2 + U(a)

1/2 m * k a^2/m + k a^2/2 = 1/2 mv^2

3/2 k a^2 = 1/2 mv^2

3 k/m a^2 = v^2

v = sqrt(3k/m) a

sqrt(3) as fast

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