The position function of a particle is given by s=t^3-3t^2-6t, t > =0.
when does the particle reach a velocity of 66 m/s?
just plug and chug:
66 = 3t^2 - 6t - 6
solve for t
To find the time when the particle reaches a velocity of 66 m/s, we can start by finding the derivative of the position function to get the velocity function.
Step 1: Find the derivative of the position function.
s = t^3 - 3t^2 - 6t
Differentiating both sides of the equation with respect to t:
ds/dt = d/dt (t^3 - 3t^2 - 6t)
= 3t^2 - 6t - 6
The derivative of the position function gives us the velocity function:
v = ds/dt = 3t^2 - 6t - 6
Step 2: Set the velocity function equal to 66 m/s and solve for t.
66 = 3t^2 - 6t - 6
Let's rearrange the equation to solve for t:
3t^2 - 6t - 6 - 66 = 0
3t^2 - 6t - 72 = 0
Step 3: Solve the quadratic equation for t.
To solve the quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case:
a = 3, b = -6, c = -72
Substituting these values into the quadratic formula, we get:
t = (-(-6) ± √((-6)^2 - 4(3)(-72))) / (2(3))
= (6 ± √(36 + 864)) / 6
= (6 ± √900) / 6
= (6 ± 30) / 6
So, t = (6 + 30) / 6 = 36 / 6 = 6 or t = (6 - 30) / 6 = -24 / 6 = -4.
Since t ≥ 0, we can disregard the negative value.
Therefore, the particle reaches a velocity of 66 m/s at t = 6 seconds.
To find when the particle reaches a velocity of 66 m/s, we need to find the time values at which the particle's velocity is equal to 66 m/s.
We know that velocity is the derivative of the position function with respect to time. So, to find the velocity function, we need to differentiate the position function, s(t), with respect to t.
s(t) = t^3 - 3t^2 - 6t
To differentiate s(t) with respect to t, we use the power rule of differentiation.
ds/dt = 3t^2 - 6t - 6
Now, we have the velocity function as ds/dt = 3t^2 - 6t - 6.
To find the time values when the particle reaches a velocity of 66 m/s, we set the velocity function equal to 66 and solve for t.
3t^2 - 6t - 6 = 66
Rearranging the equation, we get:
3t^2 - 6t - 72 = 0
Now, we can solve this quadratic equation to find the values of t at which the particle has a velocity of 66 m/s. We can either factorize this equation or use the quadratic formula.
Let's use the quadratic formula to solve for t:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation 3t^2 - 6t - 72 = 0, the coefficients are a = 3, b = -6, and c = -72.
Plugging these values into the quadratic formula:
t = (-(-6) ± √((-6)^2 - 4*3*(-72))) / (2*3)
t = (6 ± √(36 + 864)) / 6
t = (6 ± √900) / 6
t = (6 ± 30) / 6
Now we have two possible solutions for t:
t1 = (6 + 30) / 6 = 6
t2 = (6 - 30) / 6 = -4
Since time cannot be negative (as given in the initial condition t >= 0), we discard t2 = -4.
Therefore, the particle reaches a velocity of 66 m/s at t = 6 seconds.