Final answer:
The velocity v(t) is given by the derivative of the position function s with respect to time, resulting in v(t) = 3t² - 12. The acceleration a(t) is the derivative of velocity with respect to time, giving us a(t) = 6t. At t = 5 seconds, the particle's acceleration is 30 m/s².
Step-by-step explanation:
The student has provided the equation of motion for a particle, s=t³−12t, and is looking to find the velocity and acceleration as functions of time (t), as well as the acceleration after 5 seconds.
Finding the Velocity and Acceleration
To find the velocity (v(t)), we take the derivative of the position s with respect to time t:
v(t) = ds/dt = 3t² - 12
Acceleration (a(t)) is the derivative of velocity with respect to time:
a(t) = dv/dt = 6t
Finding Acceleration after 5 Seconds
To find the acceleration at t = 5 seconds, simply substitute t with 5 into the acceleration function:
a(5) = 6(5) = 30 m/s²
Therefore, the acceleration of the particle after 5 seconds is 30 m/s².