Let s(t) = 4t^3 + 48t^2 + 180t be the equation of motion for a particle. Find a function for the velocity.
v(t)=
Where does the velocity equal zero? [Hint: factor out the GCF.]
t= and t=
Find a function for the acceleration of the particle.
a(t) =
oobleck just did this for you, when you were Juan.
Juan typed it wrong, we have the same hw, i know him.
anything
To find the velocity function, we take the derivative of the equation of motion with respect to time (t).
Given s(t) = 4t^3 + 48t^2 + 180t, we can find v(t) by taking the derivative:
v(t) = d/dt (4t^3 + 48t^2 + 180t)
To differentiate the equation, we apply the power rule, which states that the derivative of t^n is n * t^(n-1). Taking the derivative term by term, we get:
v(t) = 12t^2 + 96t + 180
Therefore, the function for velocity is v(t) = 12t^2 + 96t + 180.
To find when the velocity is zero, we need to solve the equation v(t) = 0.
Setting the velocity equation to zero, we get:
0 = 12t^2 + 96t + 180
This is a quadratic equation, so we can use factoring, completing the square, or the quadratic formula to find the solutions. Since the equation does not factor easily, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
Here, a = 12, b = 96, and c = 180. Substituting these values into the formula, we get:
t = (-96 ± √(96^2 - 4 * 12 * 180)) / (2 * 12)
Simplifying further, we have:
t = (-96 ± √(9216 - 8640)) / 24
t = (-96 ± √(576)) / 24
Since 576 is a perfect square, its square root is 24. Therefore, we get:
t = (-96 ± 24) / 24
Simplifying this gives us two possible solutions:
t = (-96 + 24) / 24 = -72 / 24 = -3
t = (-96 - 24) / 24 = -120 / 24 = -5
So, the velocity is zero at t = -3 and t = -5.
To find the function for acceleration (a(t)), we need to take the derivative of the velocity equation with respect to time (t).
Taking the derivative of v(t) = 12t^2 + 96t + 180, we get:
a(t) = d/dt (12t^2 + 96t + 180)
Using the power rule again, we differentiate each term:
a(t) = 24t + 96
Therefore, the function for acceleration is a(t) = 24t + 96.