If a body is projected from the ground at the angle of 30 degrees to the horizontal with the initial velocity of 20 m/s, what maximum height and range is it going to reach?
It is possible to solve this using equations for projectile motion that can be found in the majority of formula booklets. However, since they are not always available, it is good to understand where they come from and how to derive them. In order to do so, it is best to use fundamental equations of motion ie. s(t) = s(initial) + v(initial) * t + (a * t2) / 2 for the motion in both horizontal and vertical directions. We shall assume that initial displacement is zero for both directions, there is no acceleration in x-direction and the acceleration in y is equal to gravitational acceleration and is directed downwards. v(x) and v(y) are given by 20cos(30) and 20sin(30) respectively. In order to find the maximum height, we have to determine the maximum value of y-coordinate. Taking a derivative and checking for which value of t it is equal to zero might be useful. Then, we simpy substitute this value into the original equation and easily get the height. When it comes to range, we shall find at what time the body hits the ground, ie y(t) = 0. Therefore, we solve the polynomial in y(t), take the non-zero solution and this time substitute it into the x(t) equation. It will give us the x-coordinate of the body when it hits the ground (aka range of the projectile)
