Answer: v(5) = 20.8 m/s
Step-by-step explanation:
To solve this problem, we need to use the equation of motion for the particles in the x-direction:
F = ma
where F is the net force acting on the particles, m is the mass of the particles, and a is the acceleration of the particles.
The force acting on the particles is the tension force, TE, which is given as TE = 3t² + 4 N. Since the particles are connected by the rope, they all experience the same tension force.
The mass of each particle is 3 kg, so the total mass of the system is 12 kg.
We can now find the acceleration of the particles:
F = ma
TE = ma
a = TE/m
a = (3t² + 4 N)/(12 kg)
a = (1/4)t² + (1/3) m/s²
To find the velocity of the particles after 5 seconds, we need to integrate the acceleration with respect to time:
v = ∫a dt
v = ∫[(1/4)t² + (1/3)] dt
v = (1/12)t³ + (1/3)t + C
where C is the constant of integration. To find C, we can use the initial condition that the particles are at rest at t = 0:
v(0) = 0
C = 0
Thus, the velocity of the particles after 5 seconds is:
v(5) = (1/12)(5)³ + (1/3)(5) m/s
v(5) = 20.8 m/s