Evaluate the final kinetic energy of the supply spacecraft for the actual tractor beam force, F(x)=αx3+βF(x)=αx3+β. α=6.1×10^−9 N/m^3
β=−4.1×10^6 N

K = 1.525 10⁻⁹ x⁴ + 4.1 10⁶ x

Explanation:

To find the variation of kinetic energy, let's use the work energy theorem

            W = ΔK

           ∫ F .dx = K -K₀

If the body starts from rest K₀ = 0

           ∫ F dx cos θ = K

Since the force and displacement are in the same direction, the angle is zero, so the cosine is 1

we substitute  and integrate

          α ∫ x³ dx + β ∫ dx = K

          α x⁴ / 4 + β x / 1 = K

we evaluate from the lower limit F = 0 to the upper limit F

         α (x⁴ / 4 -0) + β (x -0) = K

        K = αX⁴ / 4 + β x

        K = 1.525 10⁻⁹ x⁴ + 4.1 10⁶ x

in order to finish the calculation we must know the displacement

you get 2.3=1.21*\frac{a}2 or 4.6=1.21a or a=3.802 (b)

explanation: