Question

A single force F(x) = −8.0x (in newtons) acts on a 3.0-kg body. When x = 6.75 m, the speed of the body is 4.0 m/s. What is its speed at x = 2.5 m?

Answer 1

The potential energy of the system

$$V(x)=-\int F(x)dx=\int 8.0x dx=4.0x^2$$

The law of conservation if energy says

$$\frac{mv_1^2}{2}+V(x_1)=\frac{mv_2^2}{2}+V(x_2)$$

Hence,

$$v_2=\sqrt{v_1^2+\frac{2}{m}(V(x_1)-V(x_2))}$$

$$=\sqrt{4.0^2+\frac{2}{3.0}(4.0*6.75^2-4.0*2.5^2)}=11\:\rm m/s$$