# Mathematics Lesson

For the example of a ball being thrown up into the sky and then landing on the ground, we can model a quadratic equation to show the path of the projectile at various points in time (projectile motion). That is to say, each point plotted on the graph (parabola) will be a measurement to this effect: Suppose a ball is thrown into the sky at a velocity of 64ft/sec from an initial height of 100ft. We would set the quadratic equation as (s)0=-gt^2+v0t+h0 and substitue values for gravity, velocity, and initial height to equal 0=-16t^2+64t+100.

If we want to find out after how many seconds the ball will land, we can leave the equation set to zero and solve for t, using the quadratic formula. This will give us a solutions for t = SQRT(41)/2 (approximately 3. 2 seconds) or t = -SQRT(41)/2 (approximately -3. 2 seconds). Because we are only interested in positive values and negative values would not make any sense in this application, the ball will land after 3. 2 seconds have passed, meaning that t = +3. seconds when the postion is at 0 or ground level; Position (3. 2,0) on the Cartesian plane. Also, knowing how to quickly calculate the vertex of such models as these comes in handy. It can give us the projectile’s (ball’s) maximum height value in this equation before it begins to descend. This is simply done by finding the value of h for the x-coordinate of the parabola, and then substituting that value into the equation and solving for k, whereas h is equal to –b/2a in the standard ax^2+bx+c quadratic equation.

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Therefore, in our case, -64/2(-16)=2 is equal to the x-coordinate of the parabola’s maximum value or vertex. Substituting 2 into -16t^2+64t+100 will give us -16(2)^2+64(2)+100=164, which is the vertex’s y-coordinate. Combining them gives the maximum value of the ball’s flight; the height at 164 feet and the time at 2 seconds, which combine at the vertex point (2,164) on the Cartesian plane. Changing the quadratic equation to a quadratic function, we can see that time is a function of height, s(t)=-16t^2+64t+100, and that he correlation is that for every point in the domain of seconds there can exist only one point in the range of height. We can arbitrarily choose values from 0 to 300 (range) for height and accurately predict the time at which the ball will reach this height. Likewise we can arbitrarily chooose values for seconds from 0 to 3. 2 (domain) to find the ball’s height. Now applying a quadratic equation or function to model the flight path of a ball may sound a bit silly.

However, this is not so different from modeling the flight path of a rocket, trajectories of artillery, airplane flight paths, and planetary orbits around stars. Such applications benefit mankind (and womankind) by furthering their knowledge of science, bringing new and innovative ideas and ways of thinking. They help make our world more efficient at times, keep us safe at other times, and introduce to us exciting new products, as such would not be possible without seemingly classroom-only quadratic equations/functions, and mathematics in general.