The displacement (in meters) of a particle moving in a straight line is given by the equation of motion

s = 6/t2,
where t is measured in seconds. Find the velocity of the particle at times
t = a, t = 1, t = 2, and t = 3.

Well, to find the velocity of the particle at different times, we need to differentiate the displacement equation with respect to time. So let's do some math now.

The given equation is s = 6/t^2. When we differentiate this equation with respect to time, we get:

ds/dt = -12/t^3

Now we can substitute different values of t into this equation to find the velocity at each time.

For t = a, we have:

v(a) = -12/a^3

For t = 1, we have:

v(1) = -12/1^3 = -12

For t = 2, we have:

v(2) = -12/2^3 = -12/8 = -3/2

Finally, for t = 3, we have:

v(3) = -12/3^3 = -12/27 = -4/9

So the velocity of the particle at times t = a, t = 1, t = 2, and t = 3 is:

v(a) = -12/a^3
v(1) = -12
v(2) = -3/2
v(3) = -4/9

But remember, I'm just a Clown Bot. Take my answers with a grain of laughter!

To find the velocity of the particle at different times, we need to differentiate the equation of motion with respect to time (t) to get the velocity equation.

Given: s = 6/t^2

To find the velocity, we need to take the derivative of s with respect to t (ds/dt).

Let's differentiate the equation s with respect to t:
ds/dt = d(6/t^2)/dt

To find the derivative, we need to use the power rule of differentiation, which states that if we have a function of the form f(x) = k/x^n, the derivative is given by the formula f'(x) = -kn/x^(n+1).

Applying the power rule to the equation, we have:
ds/dt = -6(2)/t^(2+1)
ds/dt = -12/t^3

Now we have the velocity equation:
v = -12/t^3

To find the velocity at specific times, we substitute the given values of t into the equation for v.

For t = a:
v(a) = -12/a^3

For t = 1:
v(1) = -12/1^3
v(1) = -12/1
v(1) = -12

For t = 2:
v(2) = -12/2^3
v(2) = -12/8
v(2) = -3/2 or -1.5

For t = 3:
v(3) = -12/3^3
v(3) = -12/27
v(3) ≈ -0.444

So, the velocity of the particle at times t = a, t = 1, t = 2, and t = 3 are:
v(a) = -12/a^3
v(1) = -12
v(2) = -1.5
v(3) ≈ -0.444 meters per second.

you know that at any time t, the velocity

v = ds/dt = -12/t^3
so, plug in your values