A particle moves on the positive x axis (x > 0). A force F(x) = + k / x2 pushes

Question

A particle moves on the positive x axis (x > 0). A force F(x) = + k / x2 pushes the particle to larger x. Note that the force decreases to 0 as x approaches infinity. Suppose the initial conditions at t = 0 are x(0) = x0 and (dx/dt)0 = 0. Solve for the motion of the particle.

(A) Calculate the velocity in the limit that t approaches infinity.
[Data: m = 0.57 kg; k = 1.40 Nm2; x0 = 1.3 m ]

(B) Calculate the time t when v is equal to 0.50 v(?).
[Hint: The position (x) and time (t) can be related by parametric equations
x = x0 cosh2? = (x0/2){ cosh2?+ 1 }
and t = C { sinh2? + 2?}
where C is a constant, which you will need to determine.]

Explanation / Answer

initial velocity is zero
f(x)=k/x^2
a =1.4/0.57 /x^2 =2.456 /x^2
v(x) =2.456dt/x^2.......1

x =0.5*2.456 /x^2 t^2

x^3 =1.228 t^2

hece t =0.9024 x^(1.5)

dt =1.3536x^(-0.5) dx

hence equation one becomes

V(x) =2.456*dt =2.456*1.3536x^(-0.5) dx/x^2

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