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Using Quadratic Equations to Represent Motion

Author: Sophia
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what's covered
In this lesson, you will learn how to solve a quadratic equation that represents the path of an object in motion. Specifically, this lesson will cover:

Table of Contents

1. The Path of an Object in Motion

Quadratic equations can be used to represent the path of objects as they go up in the air and come back down due to the force of gravity.

EXAMPLE

A volleyball is being shot over a net, and at some distance from the net, it hits the ground. This path can be described by the following equation:

y equals short dash 0.05 x squared plus 0.3 x plus 8

In this equation, y equals the height of the ball, and x is the distance from the net. The graph below shows this relationship:



We can imagine the x-axis of the graph as the ground, the y-axis of the graph as the volleyball net, and the curve itself is the path of the volley.

We would like to know at what distance the volleyball hits the ground. To answer this question, we need to find the value of x when y equals zero. There are a couple of methods for solving quadratic equations set equal to zero. One method is to factor the equation, to get an equation in the form 0 equals open parentheses x plus a close parentheses open parentheses x plus b close parentheses. Another method is to use the quadratic formula, which uses the coefficients of the equation in standard form (a comma b, and c). We will use the quadratic formula to solve this equation, because by the looks of the coefficients in the equation, factoring is either going to be difficult or impossible.

First, identify the coefficients a comma b, and c:

y equals short dash 0.05 x squared plus 0.3 x plus 8 Identify coefficients a comma b, and c
a equals short dash 0.05 comma space b equals 0.3 comma space c equals 8 Plug coefficients into quadratic formula

Now we can use the quadratic formula to find the solutions.

x equals fraction numerator short dash b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction Plug a equals short dash 0.05 comma space b equals 0.3 comma space c equals 8 into quadratic formula
x equals fraction numerator short dash 0.3 plus-or-minus square root of 0.3 squared minus 4 open parentheses short dash 0.05 close parentheses open parentheses 8 close parentheses end root over denominator 2 open parentheses short dash 0.05 close parentheses end fraction Square 0.3 and multiply 4, -0.5, and 8
x equals fraction numerator short dash 0.3 plus-or-minus square root of 0.09 minus open parentheses short dash 1.6 close parentheses end root over denominator 2 open parentheses short dash 0.05 close parentheses end fraction Evaluate radical
x equals fraction numerator short dash 0.3 plus-or-minus square root of 1.69 end root over denominator 2 open parentheses short dash 0.05 close parentheses end fraction Simplify radical
x equals fraction numerator short dash 0.3 plus-or-minus 1.3 over denominator 2 open parentheses short dash 0.05 close parentheses end fraction Evaluate denominator
x equals fraction numerator short dash 0.3 plus-or-minus 1.3 over denominator short dash 0.1 end fraction Create two equations, one with addition and one with subtraction
x equals fraction numerator short dash 0.3 minus 1.3 over denominator short dash 0.1 end fraction comma space space space x equals fraction numerator begin display style short dash 0.3 plus 1.3 end style over denominator begin display style short dash 0.1 end style end fraction Simplify numerator in both fractions
x equals fraction numerator short dash 1.6 over denominator short dash 0.1 end fraction comma space space space x equals fraction numerator begin display style 1 end style over denominator begin display style short dash 0.1 end style end fraction Evaluate both fractions
x equals 16 comma space space x equals short dash 10 Our solutions

We have two solutions here, one which is positive and one which is negative. Looking back at our diagram, we see that taking the negative answer doesn't make sense in this context. For this reason, we'll only take the positive value, which describes the distance from the net where the volleyball hits the ground.

We can say that the volleyball will hit the ground 16 feet from the net.

hint
Always be sure to keep the context of the problem in mind. In the example above, we did not accept the negative solution, because negative distance didn't make sense in the context.

2. Calculating Maximum Height of an Object in Motion

We can also use quadratic equations to calculate the maximum height of an object that is in motion.

EXAMPLE

Think about a volleyball being shot in the air in a vertical motion, once again going up, and eventually coming back down to the ground. This relationship can be shown with the following quadratic equation:

y equals short dash 16 x squared plus 80 x plus 6

In this equation, y equals the height of the ball, and x is time. The graph below shows this relationship between the height of the ball in relationship to time:



We see that the x-axis represents time, measured in seconds, and the y-axis represents height, measured in feet.

The coefficients in this equation have important meanings within the context. short dash 16 x squaredcomes from the force of gravity on Earth, 80x represents the velocity of the volleyball (in this case, it is 80 ft/s), and 6 represents the height of the volleyball player hitting the ball.

We can use this quadratic equation to answer questions about the maximum height that the ball will reach before it begins to come back down to the ground. Looking back at our graph, we notice that the maximum height of the volleyball is at the vertex of the parabola. How can we algebraically determine the coordinates of a vertex?

We use the following equation to find the x-coordinate of the vertex (which is also the equation to the axis of symmetry of the parabola):

x space equals space minus fraction numerator b over denominator 2 a end fraction, where a and b come from coefficients of the equation in standard form

First, identify the coefficients a and b in the original equation:

y equals short dash 16 x squared plus 80 x plus 6 Identify coefficients a comma b, and c
a equals short dash 16 comma space b equals 80 comma space c equals 6 Plug into formula for x-coordinate of vertex

Now we can find the x-coordinate of the vertex.

x equals short dash fraction numerator b over denominator 2 a end fraction Plug a equals short dash 16 comma space b equals 80 comma space c equals 6 into formula
x equals short dash fraction numerator 80 over denominator 2 open parentheses short dash 16 close parentheses end fraction Evaluate denominator
x equals short dash fraction numerator 80 over denominator short dash 32 end fraction Divide 80 by -32
x equals short dash open parentheses short dash 2.5 close parentheses Simplify
x equals 2.5 x-coordinate of vertex

Our solved value for x represents the time at which the ball will reach the maximum height. We would like to calculate the height. To do this, we must plug this value of x back into the equation and find the associated y-value.

y equals short dash 16 x squared plus 80 x plus 6 Plug 2.5 in for x
y equals short dash 16 open parentheses 2.5 close parentheses squared plus 80 open parentheses 2.5 close parentheses plus 6 Evaluate
y equals short dash 100 plus 200 plus 6 Simplify
y equals 106 Our solution

This means that after 2.5 seconds, the volleyball will reach a maximum height of 106 feet.

summary
A quadratic equation can be used to model the path of an object in motion, rising and falling due to gravity. If x represents time and y represents height, the quadratic equation and/or its graph can be used to answer questions about the object's path such as the time to travel a certain horizontal distance, or the height of the object at a certain time. We can use this to calculate maximum height of an object in motion by finding the vertex.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

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