Final answer:
The particle reaches its minimum velocity when the acceleration changes from negative to positive. This can be found by solving for t when the acceleration is zero.
Step-by-step explanation:
The particle reaches its minimum velocity when the velocity is at its lowest point and starts to increase again. This occurs when the acceleration changes from negative to positive, indicating a change in direction. The minimum velocity can be determined by finding the value of t when the acceleration is zero. The particle will then start to accelerate in the opposite direction.
- Identify the equation for velocity: v(t) = 3t^3 - 842t + 12 m/s
- Differentiate the equation to find the acceleration: a(t) = 9t^2 - 842
- Solve for t when a(t) = 0: 9t^2 - 842 = 0
- Use the quadratic formula to solve for t: t = (√(842))/3 or t = - (√(842))/3
The positive value of t (√(842))/3 represents the time at which the particle reaches its minimum velocity. Substitute this value back into the velocity equation to find the minimum velocity.