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 B435U(x) = k(a2?x2)/2 for x ? a U(x)=0 for x>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). (B) Calculate the time ta when m passes the point x = a. compare ta and a/v(0).

Explanation / Answer

Since the force is conservative. Loss of Potential energy is compensated by gain in Kinetic energy.

At point x=0 Total Energy = (1/2)K*a*a + (1/2)*m*v(0)*v(0) [P.E+K.E]
At point x=a, P.E=0 and total energy is K.E which is (1/2)*m*v(a)^2

A) Equating both you can find relation between v(a) and v(0)

B) Force acting on particle is negative partial derivative of U(x) w.r.t x
therefore F = kx for 0<x<a
then acceleration = (k/m)*x
you have acceleration , distance x=a and initial velocity.. you can calculate the time it takes to travel that distance.

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