A student throws a rubber ball upward from a height of 12 feet,initially at 8 feet per second.
a.Write the position function that models the height of the ball t seconds after it was thrown.
b.What is the maximum height of the ball?
this is the "free fall" equation
a. h = -1/2 g t^2 + Vo t + Ho = -16 t^2 + 8 t + 12
b. the max height occurs on the axis of symmetry of the parabola
... tmax = -b / 2a
you can also set the 1st derivative equal to zero to find tmax
To address these questions, we can use the principles of kinematics and the equation for position.
a. To write the position function that models the height of the ball, we need to consider the initial height and the initial velocity. The equation for position is given by:
s(t) = s₀ + v₀t + (1/2)at²
Where:
- s(t) represents the position (or height) of the ball at time t
- s₀ represents the initial height of the ball
- v₀ represents the initial velocity of the ball
- a represents the acceleration (which in this case is the acceleration due to gravity, approximately -32.2 feet per second squared)
- t represents time
In this case, the initial height (s₀) is 12 feet and the initial velocity (v₀) is 8 feet per second. Therefore, the position function can be written as:
s(t) = 12 + 8t - (1/2)(32.2)t²
b. To determine the maximum height of the ball, we need to find the vertex of the quadratic function representing the height of the ball. The vertex of a quadratic function in the form of f(x) = ax² + bx + c can be found using the formula:
x_vertex = -b / (2a)
y_vertex = f(x_vertex)
In the case of the height function s(t), the vertex will represent the maximum height of the ball. By comparing the equation s(t) to the standard quadratic equation form, we can determine the values of a, b, and c:
s(t) = -16.1t² + 8t + 12
From the equation, we can see that a = -16.1, b = 8, and c = 12. Substituting these values into the vertex formula, we can calculate the maximum height:
t_vertex = -8 / (2 * -16.1)
t_vertex = -8 / (-32.2)
t_vertex = 0.25
Substituting t_vertex back into the position function s(t), we can find the maximum height of the ball:
s(t_vertex) = 12 + 8(0.25) - (1/2)(32.2)(0.25)²
Simplifying the equation, we get:
s(t_vertex) = 13.05
Therefore, the maximum height of the ball is approximately 13.05 feet.