The velocity of an object is v(t) = 27 − t2, 0 ≤ t ≤ 6 where v is measured in meters per second and t is the time in seconds.

a. Find the velocity v(2) and acceleration a(2) of the object when t = 2.
b. What can be said about the speed of the object when the velocity and acceleration have opposite signs?

(a) 23 m/s, -4 m/s²

(b) Speed is decreasing.

Explanation:

The velocity of the object is given as:

v(t) = 27 - t²

When t = 2 secs, velocity, v(2) becomes:

v(2) = 27 - (2)²

v(2) = 27 - 4

v(2) = 23 m/s

The acceleration is the first derivative of the velocity, dv/dt:

a(t) = dv(t)/dt = -2t

Acceleration after 2 secs, a(2) is:

a(2) = -2*2

a(2) = -4 m/s²

(b) When velocity and acceleration have opposite signs, it means that the velocity and acceleration are in opposite directions, hence, the object is slowing down.

In other words, the speed is decreasing.

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