Example Card: Application: Projectile Motion
Apply the quadratic discriminant to projectile motion problems in Grade 9 Algebra. Determine if a rocket reaches a target height by analyzing real, repeated, or no real solutions.
Key Concepts
Can a bit of math predict a rocket's journey? Let's use the discriminant, a key idea from this lesson, to see if a flight plan is even possible.
Example Problem A rocket is launched using the model $h = 16t^2 + 90t + 4$. Will it ever reach a height of 150 feet?
Step by Step 1. To test if the rocket reaches 150 feet, we set $h=150$ in the equation. $$ 150 = 16t^2 + 90t + 4 $$ 2. Next, we need to arrange this into standard quadratic form ($at^2+bt+c=0$). Subtract 150 from both sides. $$ 0 = 16t^2 + 90t 146 $$ 3. From this equation, we can identify our coefficients: $a= 16$, $b=90$, and $c= 146$. 4. Let's calculate the discriminant, $b^2 4ac$, to see if there's a real time $t$ when the height is 150. $$ (90)^2 4( 16)( 146) $$ 5. Performing the calculation: $$ 8100 9344 = 1244 $$ 6. The discriminant is negative. A negative result means there are no real solutions for $t$. The rocket will not reach a height of 150 feet.
Common Questions
How is the discriminant used in projectile motion problems?
The discriminant b² - 4ac from the quadratic formula tells you how many times the projectile reaches a given height. A positive discriminant means two times, zero means exactly once at the peak, and a negative means the height is never reached.
What is a double root in the context of projectile motion?
A double root occurs when the discriminant equals zero, meaning the quadratic touches the x-axis exactly once. For a projectile, it means the rocket reaches a specific height at one unique moment—typically at the apex of the trajectory.
How do you set up a quadratic equation from a projectile motion scenario?
Set the height equation equal to the target height, move all terms to one side to get ax² + bx + c = 0, then use the discriminant or quadratic formula to find the time values when that height is reached.