Suppose a coin is dropped from the top of the Empire State building in New York, which is 1,454 feet tall. The position function for free-falling objects is:
s(t) = −16t^2 + v0t + s0
where v0 is the initial velocity and s0 is the initial position.
Find the instantaneous velocities when t =1 and t = 3.
What is the name of the theorem that says there must be at least one solution to
part D?
(I already figured out C, just can't find the theorem)
what the heck is part D?
You are probably looking for the
Intermediate Value Theorem
or the
Mean Value Theorem.
To find the instantaneous velocities when t = 1 and t = 3, we need to find the derivative of the position function with respect to time. Taking the derivative of s(t) will give us the velocity function.
Given: s(t) = -16t^2 + v0t + s0
Differentiating s(t) with respect to t:
s'(t) = -32t + v0
When t = 1:
s'(1) = -32(1) + v0 = -32 + v0
When t = 3:
s'(3) = -32(3) + v0 = -96 + v0
So the instantaneous velocities when t = 1 and t = 3 are -32 + v0 and -96 + v0 respectively, where v0 is the initial velocity.
Regarding the second part of your question, the theorem that states there must be at least one solution to finding the instantaneous velocity is called the Mean Value Theorem for Derivatives.
To find the instantaneous velocities at t = 1 and t = 3, we need to differentiate the position function with respect to time, which will give us the velocity function.
Given: s(t) = -16t^2 + v0t + s0
Differentiating s(t) with respect to t:
s'(t) = -32t + v0
Now we can plug in t = 1 into the velocity function to find the instantaneous velocity at t = 1:
s'(1) = -32(1) + v0 = -32 + v0
Similarly, plugging in t = 3:
s'(3) = -32(3) + v0 = -96 + v0
So the instantaneous velocity at t = 1 is -32 + v0, and at t = 3 is -96 + v0.
As for the theorem that states there must be at least one solution to part D, the theorem is called the Intermediate Value Theorem (IVT). The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], then it takes on every value between f(a) and f(b) at least once.
In this context, if we have a continuous function of time representing the position and velocity of the falling coin, and we know that the coin starts at the top of the Empire State building (initial position) and falls to the ground (final position), the Intermediate Value Theorem guarantees that at some point during its fall, the instantaneous velocity will be zero.