Question 5

The force $$\vec{F} = F\hat{i}$$ on a particle of mass 2 kg, moving along the $$x$$-axis is given in the figure as a function of its position $$x$$. The particle is moving with a velocity of 5 m/s along the $$x$$-axis at $$x = 0$$. What is the kinetic energy of the particle at $$x = 8$$ m?

Given: $$m = 2\text{ kg}$$, $$v_i = 5\text{ m/s}$$

$$K_i = \frac{1}{2}mv_i^2 = \frac{1}{2}(2)(5)^2 = 25\text{ J}$$

Using area under the curve:

$$W = \text{Area}_{0 \to 2} + \text{Area}_{2 \to 5} + \text{Area}_{5 \to 8}$$

$$W = \frac{1}{2}(2)(2) - \frac{1}{2}(1 + 3)(1) + \frac{1}{2}(3)(3) = 2 - 2 + 4.5 = 4.5\text{ J}$$

From Work-Energy Theorem: $$K_f = K_i + W = 25 + 4.5 = 29.5\text{ J}$$

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