An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
First student's height: y = − 16 t 2 + 15 t + 3 .
Second student's height: y = − 16 t 2 + 35 t + 4 .
Setting the equations equal: − 16 t 2 + 15 t + 3 = − 16 t 2 + 35 t + 4 .
Solving for t yields t = − 20 1 , which is not a valid time. Therefore, the solution is no solution because time is negative.
Explanation
Problem Setup We are given the equation for projectile motion: y = − 16 t 2 + v 0 t + h 0 , where y is the height, t is the time, v 0 is the initial upward velocity, and h 0 is the initial height. We have two students throwing tennis balls. The first student has v 0 = 15 ft/s and h 0 = 3 ft. The second student has v 0 = 35 ft/s and h 0 = 4 ft. We need to write a system of equations to represent this situation, find the solution to the system, and analyze the solution.
Writing the Equations First, let's write the equations for each student's tennis ball trajectory:
First student: y = − 16 t 2 + 15 t + 3
Second student: y = − 16 t 2 + 35 t + 4
Setting the Equations Equal To find the solution to the system, we set the two equations equal to each other: − 16 t 2 + 15 t + 3 = − 16 t 2 + 35 t + 4
Solving for Time Now, let's solve for t :
15 t + 3 = 35 t + 4 0 = 20 t + 1 t = − 20 1 Since time cannot be negative, this solution doesn't make sense in the context of the problem.
Analyzing the Solution Since the time t cannot be negative, there is no point of intersection between the two trajectories in the physical world (where time is non-negative). Therefore, there is no real solution to this system of equations.
Final Answer The solution to this system of nonlinear equations is no solution because it results in a negative value of time.
Examples
Understanding projectile motion is crucial in sports like baseball or basketball, where players need to calculate the trajectory of a ball to make accurate throws or shots. By using equations that model projectile motion, athletes and coaches can optimize their techniques and strategies. For example, knowing the initial velocity and height at which a ball is thrown allows one to predict its range and maximum height, which can inform decisions about the best angle and force to use for a particular throw.
The total charge that flows through the device is 450 coulombs, and this corresponds to approximately 2.81 x 10^21 electrons. This is calculated using the relationship between current, time, and the charge of a single electron. Therefore, a significant number of electrons flow through when the device operates for 30 seconds at 15.0 A.
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