[meekadn]

How far will a stone travel over level ground if it is thrown upward at an angle of 30 degrees with respect to the horizontal and with a speed of 30 m/s?

What is the maximum range that can be achieved if a stone is thrown with an initial speed of 24 m/s?

[dmatos]

First thing to do is break the 30m/s into a horizontal and a vertical velocity. Use trigonometry for this (Vy = 30m/s*sin(30deg), Vx = 30m/s*cos(30deg)). Now, throughout the flight, the horizontal velocity will remain unchanged. The vertical velocity will decrease due to gravity, and then increase on the way back down. When it hits the ground, it will be travelling at -Vy (if there is no air friction).

So, how long is the rock in the air? The amount of time it takes acceleration due to gravity (9.8m/s^2) to change Vy into -Vy:

Vfinal = Vinitial * 9.8t

solve for t.

Then, how far did it travel horizontally? Well, it was in the air for time t, and:

d = Vx*t

As for the second part of the question, the variable you have is theta, the angle at which it is thrown. Solve the question again, but in this case, solve it analytically so that your final answer has the variable theta in it. Then, find the maximum of that equation for theta between 0 and 90 degrees. To find a maximum, take the derivative and then set that equal to zero, of course. Make sure you check that it's a max and not a min!