The force acting on a stationary particle of mass 5 kg varies with displacement as given. The change in velocity of the particle during the motion.

Steps to Solve

1. Determine the Force Function The force $F$ acting on the particle varies with displacement $x$ as shown in the graph. We can identify the piece-wise force function based on the provided triangle. The vertices of the triangle are at points $(0,0)$, $(10,20)$, and $(10,0)$.

2. Calculate the Work Done To find the change in velocity, we need to calculate the work done on the particle. The work done, $W$, is given by the area under the force-displacement graph:

   • For the triangle, the area can be calculated as: $$ W = \frac{1}{2} \times \text{base} \times \text{height} $$

   Here, the base is 10 m and the height is 20 N. Thus, $$ W = \frac{1}{2} \times 10 \times 20 $$

3. Evaluate the Work Done Calculating the work done: $$ W = \frac{1}{2} \times 10 \times 20 = 100 \text{ J} $$

4. Use the Work-Energy Principle According to the work-energy principle, the work done on an object is equal to the change in kinetic energy: $$ W = \Delta KE = \frac{1}{2} m (v^2 - u^2) $$ Since the particle starts from rest, $u = 0$. Thus, $$ 100 = \frac{1}{2} \times 5 \times v^2 $$

5. Solve for Velocity Rearranging the equation gives: $$ 100 = \frac{5}{2} v^2 $$ Multiplying both sides by 2: $$ 200 = 5v^2 $$ Now divide by 5: $$ 40 = v^2 $$

Taking the square root: $$ v = \sqrt{40} \approx 6.32 \text{ m/s} $$