Let s ( t ) = 4 t 3 + 48 t 2 + 180 t 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)=
s = 4t^3 + 48t^2 + 180t
v = 12t^2 + 96t + 180 = 12(t^2+8t+15) -- ??? this seems strange
a = 12(2t+18) = 24(t+9)
Well, it seems like we're dealing with some high-level math here! Don't worry, I'm here to make it a little more fun for you.
To find the velocity function, we need to find the derivative of the equation of motion. So let's put on our mathematical clown noses and get to work!
Taking the derivative of s(t) = 4t^3 + 48t^2 + 180t, we get v(t) = 12t^2 + 96t + 180.
Now, let's find where the velocity equals zero. This is like looking for the "parking brake" of our particle, where it temporarily stops. To find this, we can set v(t) = 0 and solve for t.
0 = 12t^2 + 96t + 180
To make things a little easier, let's try to factor out the greatest common factor (GCF) from this equation. Unfortunately, our lovely clown noses are telling us that this equation cannot be factored nicely. But fear not, brave mathematician! We can use the quadratic formula to find our solutions.
Using the quadratic formula, we have t = (-b ± √(b^2 - 4ac)) / (2a).
Plugging in the values from our equation, we get t = (-96 ± √(96^2 - 4 * 12 * 180)) / (2 * 12).
After some calculations, we find that t ≈ -5.07 and t ≈ -13.93.
And finally, let's find the function for the acceleration of the particle. We'll need to take the derivative of the velocity function, so let's put on our mathematical clown hats and get to it!
Taking the derivative of v(t) = 12t^2 + 96t + 180, we get a(t) = 24t + 96.
And there you have it! The velocity function is v(t) = 12t^2 + 96t + 180 and the acceleration function is a(t) = 24t + 96. I hope I was able to bring some laughter to your math journey!
To find the function for velocity, we need to take the derivative of the equation of motion.
Step 1: Find the derivative of s(t) with respect to t.
s'(t) = 12t^2 + 96t + 180
So, the function for velocity, v(t), is given by:
v(t) = 12t^2 + 96t + 180.
To find where the velocity equals zero, we need to solve the equation v(t) = 0.
Step 2: Set v(t) equal to zero and solve for t.
12t^2 + 96t + 180 = 0
Step 3: Factor out the greatest common factor (GCF) to simplify the equation.
12(t^2 + 8t + 15) = 0
Step 4: Factor the quadratic equation t^2 + 8t + 15.
(t + 3)(t + 5) = 0
Step 5: Set each factor equal to zero and solve for t.
t + 3 = 0 or t + 5 = 0
t = -3 or t = -5
So, the velocity equals zero at t = -3 and t = -5.
To find the function for acceleration, we need to take the derivative of the velocity function.
Step 6: Find the derivative of v(t) with respect to t.
v'(t) = 24t + 96
So, the function for acceleration, a(t), is given by:
a(t) = 24t + 96.
To find the function for the velocity, we need to differentiate the equation of motion with respect to time. The velocity function, v(t), is the derivative of the position function, s(t).
So, let's differentiate s(t) = 4t^3 + 48t^2 + 180t:
s'(t) = (d/dt)(4t^3) + (d/dt)(48t^2) + (d/dt)(180t)
Taking the derivative of each term:
s'(t) = 12t^2 + 96t + 180
Therefore, the function for the velocity, v(t), is:
v(t) = 12t^2 + 96t + 180
To find when the velocity equals zero, we need to solve the equation v(t) = 0.
Let's set the velocity function equal to zero:
12t^2 + 96t + 180 = 0
To solve this quadratic equation, we can first factor out the greatest common factor (GCF), which is 12:
12(t^2 + 8t + 15) = 0
Now, let's factor the quadratic inside the parentheses:
12(t + 3)(t + 5) = 0
Setting each factor equal to zero:
t + 3 = 0 solves for t = -3
t + 5 = 0 solves for t = -5
So, the velocity is equal to zero at t = -3 and t = -5.
To find the function for the acceleration, a(t), we need to differentiate the velocity function, v(t), with respect to time. The acceleration function is the derivative of the velocity function.
So, let's differentiate v(t) = 12t^2 + 96t + 180:
v'(t) = (d/dt)(12t^2) + (d/dt)(96t) + (d/dt)(180)
Taking the derivative of each term:
v'(t) = 24t + 96
Therefore, the function for the acceleration, a(t), is:
a(t) = 24t + 96