Final answer:
To find the velocity to hit the teacher with the metronome, we use projectile motion equations, separating horizontal and vertical components, with the vertical motion determining the time of flight and the horizontal motion requiring a constant velocity over the 10.8 m distance.
Step-by-step explanation:
The scenario described is an application of the principles of projectile motion and kinematics in physics. To calculate the necessary horizontal initial velocity at which the drum major must throw the metronome to hit the physics teacher, we analyze the motion separately in the horizontal (x) and vertical (y) directions, given that projectile motion is two-dimensional.
For the horizontal motion, since there is no air resistance, the velocity remains constant. The formula for horizontal distance traveled is d = vt, where d is distance, v is velocity, and t is time. To find the velocity, we need to find the time it takes for the metronome to hit the ground, which comes from analyzing the vertical motion.
The vertical motion can be described by the equation y = y0 + v0yt + ½at2, where y is the final vertical position (0 m when it hits the ground), y0 is the initial vertical position, v0y is the initial vertical velocity (0 m/s since the throw is horizontal), and a is the acceleration due to gravity (9.81 m/s2). Using this, we can solve for the time t it takes for the metronome to fall from the release height of the drum major's head level, which is 2.00 m (podium height) + 1.73 m (drum major height) - teacher's head level.
Once we have the time, we insert it back into the first equation to solve for the required horizontal velocity. This will be the velocity at which the drum major must throw the metronome to hit the teacher at a distance of 10.8 meters.