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## Homework Statement

A boomerang is thrown with an initial linear velocity of 5 m/s at an angle of 30 degrees vertically. The initial angular velocity is ##2\frac{revolutions}{s}## At its peak, it has a displacement about the z axis of 2 meters and about the x-axis of 10 meters. The force applied on the boomerang during the throw is 10N with no force in the z direction. It has a mass of 2 kg and a length of .2 m. How long does it take for the boomerang to return?

## Homework Equations

##F=ma##

##ω=ω_0+αt##

## The Attempt at a Solution

Okay so first off let's assume it starts at 0 on both the x and z axes. Since it goes further along the x-axis than the z axis, it must have a higher velocity in the x direction for most of its path. Next up we need to take all the forces into account. There are 2 forces imparted on the boomerang. There is the initial throwing force of 10N which becomes the torque of the boomerang. There is also the force of gravity.

The force of gravity is ##F_g=mg## according to Newton's second law. This is ##F_g=2 kg*9.8\frac{m}{s^2}## which multiplies out to give us ##F_g=19.6N##

The initial force needs to be separated into its x and y components. The x component is going to be ##F_x=10N * sin(30°) = 10N * .5 = 5N## This means that ##F_y=5N## The velocity in both the x and y directions also is split in half so each velocity initially is ##2.5\frac{m}{s}##

Now, in order to not fall and to actually come back to its original position, the boomerang has to accelerate in the direction of its motion since gravity starts out being the predominant force. It must have a rotational acceleration such that at its peak, the torque balances out with gravity. In other words as time passes, the acceleration must increase which means that the angular velocity has to change, even if the linear velocity stays constant for some time ##t##.

And this is where I'm stuck. I should have enough information here to figure out the angular acceleration and thus the time it takes for the boomerang to return but I can't seem to figure it out. Can somebody help me with this? Also I have no idea why the inline latex isn't showing up properly.

EDIT: I fixed the error with the Latex

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