A particle travels along a straight path with its position given by x = 6t² – t³, where t is in seconds and x in meters. What is the highest velocity reached by the particle during its motion?

Explanation

The position function is x = 6t² – t³. The velocity is the first derivative of position with respect to time: v = dx/dt = 12t – 3t². To find the maximum velocity, set the acceleration (the derivative of velocity) to zero: a = dv/dt = 12 – 6t = 0, which gives t = 2 seconds. Substituting back, v = 12(2) – 3(2)² = 24 – 12 = 12 m/s. Therefore, the maximum velocity is 12 meters per second.