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A Vertical Problem Example
A ball is thrown vertically upward.
After t seconds its height h (in feet) is given by the function,
h(t) = 64t - 16t2
After how long will it reach its maximum height?
Before we start working on this problem, let's see what the equation represents.
h(t) = -16t2 + 64t + 0
h(t) is the height in terms of time, t.
-16 represents the value of the force of gravity. This is standard.
You will see -16 when using feet, and -4.9 when using meters.
64 represents the upward velocity given in the problem.
0 represents the height of the projectile at the time of launch.
In this case, the ball is tossed vertically without knowing the person's height.
Questions
How do you know that it will reach the max (not min) height?
What value tells you this?
In what way does this graph open - up or down, and why?
( Think how you would know you are looking for the max and not the min)
Of course, the author does ask us to find the max - but the equation tells us more!)
Process
Algebraically
~ Identify a and b of the equation.
~ Plug-in a and b in -b/(2a) = t to find WHEN the max occurred.
~ Use the above value to determine WHAT the max value will be h(2) = 64 ft.
Graphically
~ Use Desmos, graph the function, and find its vertex.
~ To see the whole graph, change the domain and the range in Desmos.
~ After changing these values, find the t-value on the graph.
~ Click on this value to determine what the value would be if t = 2, then h(2) = 64.
Problem 1
The equation h(t) = -9.8t2 + 20t + 18 models the height of an object, in meters, above the ocean t seconds after it is shot from the deck of the ship.
a. What does 20 in the context of this function represent?
b. What does 18 in the context of this function represent?
Problem 2
A physics class is launching a model rocket from the roof of a building, 32 feet above the ground. The rocket is launched with an initial upward velocity of 84 feet per second.
a. Write an equation to model the path of the rocket in the air.
(Let h(t) represent the height of the rocket t seconds after the launch.
Problem 3
An arrow is fired from a bow, and its height, h, in meters above the ground, t seconds after being fired, is given by h(t) = -5t2 + 40 t + 3.
Algebraically determine the maximum height attained by the arrow and the time taken to reach this height.
(Use the process as in the example)
a. Find the time the arrow reaches its max height.
b. Once you find the time, find the max height.
Answers
Problem 1
a. 20 meters/second in the initial upward velocity.
b. 18 meters is the starting height of the object above the ocean (or the distance between the ocean surface and the ship's deck.
Problem 2
a. h(t) = -16t2 + 84t + 32
Problem 3
a. t = -b/(2a) = -40/[2(-5)] = 4 seconds
b. h(4) = -5(4)2 + 40(4) + 3 = 83 meters.
The arrow reaches a max height of 83 meters in 4 seconds.
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