Example
Harry decided to play a game of Quidditch with some muggles. Since the muggles didn't have any powers, they decided to play on the ground (not flying in the air). A few seconds after the Snitch is released (i.e. the Quidditch game starts), Harry throws the enchanted Quaffle (assume his throw starts from the ground) towards the hoop. He misses the hoop and the Quaffle hits the ground. The height of the Quaffle, Q, in meters after s seconds since the game started is given by
a) What was the height of the Quaffle 9 seconds after the game started?
b) How long after the Quidditch game started did Harry throw the Quaffle? How long after the Quidditch game started did the Quaffle hit the ground?
c) When will the Quaffle be at 292.8 meters high?
d) What is the maximum height of the Quaffle? How long after the game started did it reach the maximum height?
Solution
a) Q represents the height of the Quaffle in meters while s represents the time in seconds. To find the height of the Quaffle, substitute s with 9 and simplify to find Q. First set up the equation.
b) This problem asks how long after the game started did Harry throw the Quaffle, and how long until it hits the ground. In the problem it states we assume that it starts on the ground. Hence, this problem is asking to find the two values when the height is zero. Thus, to solve, we set Q = 0, and then solve for s
So in 7 seconds after the game started, Harry threw the Quaffle. In 21 seconds after the game started, the Quaffle will touch the ground.
c) To solve the problem, you must set Q=292.8, and then solve for s.
After 13 and 15 seconds the Quaffle with be at 292.7 meters high.
d) To find the maximum of height of the Quaffle, we need to find the vertex of this parabola. The maximum is the height, Q, or the y-coordinate of the vertex. Time is, s, or the x-coordinate of the vertex. If we use the formula -b over 2a, remember that this calculates the x-coordinate, and to find the y-coordinate, we have to evaluate the original function at that x-value.
We have found 14, but remember, this is the x-coordinate of the vertex, represented by the variable s in our problem, which is time in seconds, so in 14 seconds, the Quaffle will reach the maximum height. Now, to find the maximum height itself, we substitute this value for s in the original function:
298.9 meters is the maximum height of the Quaffle after 14 seconds whe nthe game started.
Equation, Table, and graph
REFLECTION
This example is about finding the distance or height if the Quaffle in a Quidditch game after seconds the game starts. What we need to find out is at what height will the Quaffle will be after a certain amount of seconds. The problems will ask how long will it be when the Quaffle will be at 300 meters or 20 seconds after the Quaffle is thrown, what height will it be. Generally speaking, projectile motion problems involve objects that are thrown, shot, or dropped. Usually the object will be launched directly upward or dropped directly down. This question asks for the time when the object strikes the ground. Another way of thinking of this is, when is the object's height zero (on the ground)? So, set s equal to zero and solve the equation. Example being, after ten seconds the Quaffle would be at 250 meters high. This word problem is fair test of mathematics since it involves quadratics, factoring, completing the square and dividing. Overall this problem is a good step to the quadratic equations after this property.