First student: [tex]y=-16 t^2+[/tex] [ ] t + [ ]
Second student: [tex]y=-16 t^2+[/tex] [ ] t + [ ]
The solution to this system of nonlinear equations is [ ] because it results in a [ ] value of time.
Answer (1)
First student's height: y = − 16 t 2 + 15 t + 3 .
Second student's height: y = − 16 t 2 + 35 t + 4 .
Set the equations equal: − 16 t 2 + 15 t + 3 = − 16 t 2 + 35 t + 4 .
Solve for t : t = − 20 1 , which is not a valid time.
The solution is no solution because it results in a negative value of time.
Explanation
Write the system of equations Let's first write the equations for the height of each tennis ball as a function of time. For the first student, the initial height h 0 = 3 feet and the initial upward velocity v 0 = 15 feet/second. Substituting these values into the general equation, we get:
y = − 16 t 2 + 15 t + 3
For the second student, the initial height h 0 = 4 feet and the initial upward velocity v 0 = 35 feet/second. Substituting these values into the general equation, we get:
y = − 16 t 2 + 35 t + 4
So the system of equations is:
y = − 16 t 2 + 15 t + 3 y = − 16 t 2 + 35 t + 4
Solve the system of equations To find the solution to this system, we need to find the time t when the heights y are equal. We can set the two equations equal to each other:
− 16 t 2 + 15 t + 3 = − 16 t 2 + 35 t + 4
Now, we can solve for t :
15 t + 3 = 35 t + 4
Subtract 15 t from both sides:
3 = 20 t + 4
Subtract 4 from both sides:
− 1 = 20 t
Divide by 20:
t = − 20 1
Since time cannot be negative, this solution is not physically meaningful. The tennis balls never reach the same height at any real time. Therefore, there is no solution to this system of equations that makes sense in the context of the problem.
Analyze the solution Since the time value is negative, it is not a valid solution in this context.
Examples
Understanding projectile motion is crucial in sports like tennis, where players need to predict where the ball will land. By modeling the ball's trajectory with a quadratic equation, players can estimate the optimal angle and velocity to hit the ball for maximum performance. This involves considering factors like initial height, launch angle, and air resistance to accurately predict the ball's path and improve their game strategy.
