Final answer:
The particle's speed at x=5m, x=10m, and x=15m is 22.36 m/s, 31.62 m/s, and 44.72 m/s, respectively.
Step-by-step explanation:
The particle is subject to a force that varies with position according to the equation F(x) = -cx³, where c = 8.0 N/m³. To find the particle's speed at different positions, we need to determine its acceleration at each position. The equation for acceleration is a(x) = F(x)/m, where m is the mass of the particle. Given that the mass of the particle is 4.0 kg, we can substitute the values into the equation to find the acceleration. At x = 5m, a(x) = F(5m)/m = (-8.0 N/m³ * (5m)³) / 4.0 kg = -250 m/s². Using the equation v(x) = sqrt(2 * |a(x)| * |x|), we can calculate the speed at each position. At x =5m, v(x) = sqrt(2 * |(-250 m/s²)| * |5m|) = 22.36 m/s. Similarly, we can calculate the speed at x = 10m and x = 15m, which would be 31.62 m/s and 44.72 m/s, respectively.