Physics Exercises/Conservation of Energy

Part 1. (Medium)
Many kinematic problems seem to be relatively simple once one knows the basic equations of motion. And conservation of energy is simple when all the equations are in place as well. The question is then on problems which deal with both. Take a simple sling, which is rotated in an axis perpendicular to that of gravity. This sling is then released, and the payload travels a distance R before landing where it is supposed to go.

For a simple sling, this is a purely kinematic problem. Gravity pulls down on the payload as it travels as follows:

$$y''(t) = -g$$ $$x''(t) = 0$$

Hence the equation for motion of each of the axis is at:

$$y(t) = -gt^2/2 + V_\mathrm{y_0}t + y_0$$ $$x(t) = V_\mathrm{x_0}t + x_0$$

And from there, one can calculate the range given:


 * $$g$$, The gravitational constant, i.e. 9.81 m/s²
 * $$x_0$$, The position of the object in the x-cordinate at time t=0
 * $$x_0$$, The position of the object in the y-cordinate at time t=0
 * $$V_\mathrm{x_0}$$, The velocity of the object in the x-cordinate at time t=0
 * $$V_\mathrm{y_0}$$, The velocity of the object in the y-cordinate at time t=0.

Hence, with this set of equations, anyone can calculate the range. Simply solve for t in the x equation, plug this value of t in the y equation. This method will work if you've got a flat surface with a function y=0, or some other function y=f(x), which is why I prefer it. Simply find the intersection of the two functions (the ground function, and the new one you created) to solve for x when the functions collide. Use this to solve for y. If the problem asks for it, use the value found for x and use it to solve for t.

So, imagine you have a ball on a string of length r, which is always traveling at a tangential velocity V. Find the range of this ball using pure kinematics, as a function of the angle released. Find the maximum range of the ball.


 * a.) Is it better to swing underhand or overhand?
 * b.) What angle gives the most range?
 * c.) What angle would make it so the ball passed right through the center of the sling circle?

Part 2. (Hard)
Turn the sling into a catapult. What's the release angle on this to make for optimum range? Note that the velocity with respect to angle is no longer constant. It will increase linearly with height above the ground, much like the descending weight's velocity will increase as it gets closer to the ground.


 * a.) What is the range of the catapult?
 * b.) What's the max range?

Part 3. (Very Hard)
Now let's beef up the scale. Back to the sling. Now the radius of the circle is so large that the potential energy difference between the bottom of the circle and the top of the circle is not negligible. Assuming this is the only change:


 * a.) What is the range of the sling, as a function of release angle?
 * b.) What angle gives the max range?
 * c.) Is it still better to swing underhand or overhand?
 * d.) What angle would make it pass through the center of the circle?

(If you get a quartic equation in cos²(θ) like I did, just post that equation.)