MAC, Inc. (the company you work for) has decided to move forwardwith a feasibility

study for a product it wishes to manufacture and sell. This productis a particle launcher that should have the capability, for everyactivation, of launching two particles into the air at twodifferent times and at two different angles of elevation, in anautomated fashion so that they both land at the same distance fromthe launch point at the same time. For example, let P1 and P2 betwo particles, let T1 and T2 be two times, and let theta1 andtheta2 be two angles of elevation. Once activated, the launchershould be able to operate so that particle P1 is launched at timeT1 with an angle of elevation theta 1, and particle P2 islaunched

at time T2 with an angle of elevation theta2.

The launcher must be configured by a technician, prior to eachactivation, so that the

initial speed is set for the launcher, and the distance from thelauncher to where both

particles will land is set. For each activation of the launcher,the initial speed will be the

same for both particles. Once the technician has set the initialspeed and desired distance,

the launcher must calculate the angles of elevation to use and howmuch time to wait in

between particle launches. Then, when the technician activates thelauncher, both

particles will be launched at the appropriate angles of elevationand at the appropriate

times so that they both land at the configured distance from thelauncher at the same time.

The launcher, based on its calculations, will change the angle ofelevation to be ready for

the second particle launch. The launcher, however, must flash awarning for the

technician if the information the technician used to configure thelauncher does not result

in a two-particle launching solution.

Certain assumptions must be made. It is assumed that that launchercan always change

the angle of elevation to get ready for the second particle launchfaster than the time

delay calculated for the second particle launch. It is assumed thatthe units for speed are

meters per second, that the units for distance are meters, and thatgravity is 9.8 meters per

second squared. It is assumed that gravity is the only force actingon the motion of the

particles after launch, and that there is no wind to be considered,so that the particles land

at a point exactly in line with the direction the launcher ispointed.

The questions that must be answered and written up in anengineering report are as

follows:

1. Given the initial speed and desired landing distance for theparticles, find the

mathematical expressions that must be used to find 1, 2, and thetime delay

between particle launches.

2. Determine under which conditions the launcher will not be ableto perform the

operation requested, and will consequently flash a warning to thetechnician.

3. Each particle launched will follow a path of motion that isparabolic. Show that

the vertex of the parabola for each particleâs path occurs at thesame distance from

the launcher.

4. Along the path of each launched particle, there will be amaximum curvature.

Find that maximum curvature for each launched particle.