To calculate the acceleration of a moving object when its velocity is given as a function of time, you can differentiate the velocity function with respect to time. Mathematically, acceleration is the rate of change of velocity with respect to time, so taking the derivative of the velocity function will give you the acceleration function.
Let's say the velocity function is denoted as v(t). To find the acceleration function a(t), you differentiate v(t) with respect to t:
a(t) = d/dt [v(t)]
For example, if the velocity function is given as v(t) = 3t² + 2t + 1, you can calculate the acceleration by differentiating this function:
a(t) = d/dt [3t² + 2t + 1]
Taking the derivative term by term:
a(t) = d/dt (3t²) + d/dt (2t) + d/dt (1)
Simplifying:
a(t) = 6t + 2
So, the acceleration function is a(t) = 6t + 2.
This acceleration function describes how the acceleration of the object changes with respect to time. To find the acceleration at a specific time t, you can substitute that value into the acceleration function. For example, if you want to find the acceleration at t = 2 seconds:
a(2) = 6(2) + 2
a(2) = 12 + 2
a(2) = 14 m/s²
Therefore, the acceleration of the object at t = 2 seconds is 14 m/s².