From here we ask which angle sends the ball farthest. The case treated is a throw from the ground that lands back on the ground.
Start with the time in the air. The question is when the ball returns to the ground, so it is a question about height. However fast the ball is travelling sideways, the moment it comes down is unaffected. The independence of the horizontal and the vertical, seen in the last lecture, is doing the work here.
The initial vertical velocity was . The ball rises, the vertical velocity reaches zero at the top, and it comes back down. The climb and the descent take the same time, so the whole flight is twice the climb.
The climb takes , so the time in the air is . The marks in the figure are positions at equal intervals of time, and they fall in the same pattern on either side of the line through the top.
Next the distance. No force acts along the horizontal, so the ball keeps travelling at . The motion is uniform, so the distance is nothing but speed times time.
That time is the hang time just found, giving . Putting in leaves .
can be rewritten as . Gathered up, the range is .
The angle appears in that equation in one place only, inside . If the initial speed and the acceleration due to gravity are fixed, then the whole of the range is settled at that single spot.
A sine never exceeds 1, whatever the angle. reaches exactly 1 when , which is to say when .
So a throw at goes farthest, and the distance is then .
The figure lays five paths over one another, apart. The bright one is , and it reaches farthest to the right. Throw flat and the ball travels quickly sideways but comes down at once. Throw steeply and it hangs a long time but makes little headway. The balance between the two is struck right in the middle.
Look closely at the landing marks, though. There are five paths, and only three marks.
There are only three marks because and land in the same place, and so do and . Each pair adds up to .
The equation confirms it. Putting in place of gives , which equals . So .
Landing in the same place, however, is not the same as landing in the same way. The two balls in the figure leave together, but the steeper one hangs in the air far longer, because the hang time depends on the angle. The flatter throw arrives first and the steeper one catches up later.
Put another way, there are two throws that hit the same target. Use the flat one to get there quickly, the steep one to clear a wall in between. Only stands alone, and it is the point where the two branches meet.