Question

The equation of motion of a particle is s=t3−12t, where s is measured in meters and t is in seconds. (Assume t≥0.) (a) Find the velocity and acceleration as functions of t. v(t)= a(t)= (b) Find the acceleration, in m/s2, after 5 seconds. m/s2 (c) Find the acceleration, in m/s2, when the velocity is 0 . m/s2

Answer 1

Answer:The acceleration when the velocity is 0 is 12 m/s^2.

Step-by-step explanation:

(a) To express the velocity and acceleration as functions of time (t) using the given equation of motion, we need to differentiate the equation with respect to time.

Given: s = t^3 - 12t

First, let's determine the velocity (v):

The derivative of s with respect to t gives us the velocity:

v(t) = ds/dt

Differentiating the equation of motion:

v(t) = d/dt(t^3 - 12t)

= 3t^2 - 12

Therefore, the velocity as a function of time is v(t) = 3t^2 - 12.

Next, let's calculate the acceleration (a):

The derivative of v with respect to t gives us the acceleration:

a(t) = dv/dt

Differentiating the velocity equation:

a(t) = d/dt(3t^2 - 12)

= 6t

Therefore, the acceleration as a function of time is a(t) = 6t.

(b) To determine the acceleration after 5 seconds, substitute t = 5 into the acceleration equation:

a(5) = 6(5)

= 30 m/s^2

Hence, the acceleration after 5 seconds is 30 m/s^2.

(c) To find the acceleration when the velocity is 0, set v(t) = 0 and solve for t:

3t^2 - 12 = 0

Solving this quadratic equation yields:

3t^2 = 12

t^2 = 4

t = ±2

Since we are considering t ≥ 0, we discard the negative value and select t = 2.

Substituting t = 2 into the acceleration equation:

a(2) = 6(2)

= 12 m/s^2

Therefore, the acceleration when the velocity is 0 is 12 m/s^2.