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

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 for x > a

The initial values are x(0) = 0 and v(0)=v*sqrt(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 for x > a The initial values are x(0) = 0 and v(0)=v*sqrt(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

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.

Hope this helps

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