A silver dollar is dropped from the top of a building that is 1398 feet tall. Use the position function below for free-falling objects. (Round your answers to 3 decimal places.)
s(t) = -16t2 + v0t + s0
(a) Determine the position and velocity functions for the coin.
s(t) =
v(t) =
To determine the position and velocity functions for the coin, we need to understand the given position function and the initial conditions.
The given position function for free-falling objects is:
s(t) = -16t^2 + v0t + s0
where:
- s(t) represents the position of the object at time t.
- v0 represents the initial velocity of the object.
- s0 represents the initial position of the object.
In this case, the object is a silver dollar dropped from the top of a 1398-foot building. This means that the initial position (s0) is 1398 feet, and the initial velocity (v0) is 0 since the object starts from rest.
Now, let's substitute these values into the position function:
s(t) = -16t^2 + 0t + 1398
s(t) = -16t^2 + 1398
This is the position function for the coin.
To find the velocity function (v(t)), we need to take the derivative of the position function with respect to time (t). The derivative gives us the rate of change of position with respect to time, which is the velocity.
v(t) = d/dt [-16t^2 + 1398]
v(t) = -32t
Therefore, the velocity function for the coin is v(t) = -32t.
So, the position function for the coin is s(t) = -16t^2 + 1398, and the velocity function is v(t) = -32t.
To determine the position and velocity functions for the coin, we need to consider the initial conditions.
Given:
Building height (s0) = 1398 feet
Acceleration due to gravity (g) = -32 ft/s^2 (negative due to downward direction)
We know that the coin is dropped from rest (initial velocity v0 = 0).
(a) Position function (s(t)):
s(t) = -16t^2 + v0t + s0
Plugging in the values, we have:
s(t) = -16t^2 + 0t + 1398
Therefore, the position function for the coin is:
s(t) = -16t^2 + 1398
(b) Velocity function (v(t)):
To find the velocity function, we need to take the derivative of the position function with respect to time (t).
Differentiating the position function, we get:
v(t) = -32t + 0
Since the initial velocity is zero, the velocity function for the coin is:
v(t) = -32t