This is a simple demonstration you can do at home to show the independence of vertical and horizontal motion. All you need for it is a table top and two pennies, or two coins of any kind. Place one coin on the edge of the table, and place
the other coin near the first one on the table. What you're going to do with the other coin is flip it with your finger so that it just grazes the first one and the first one will drop at the same time the second one is projected off the table.
The question is when will they hit? Which one
will hit first or will they hit at the same time? Let's go ahead and try this and see what happens. If you were listening to the sound you probably heard only a single sound when they hit they hit at the same time. Let's do that one more time.
they hit at the same time even though one coin has to travel a further distance to get to its target? Well again because the vertical and horizontal motions are independent of each other the objects fall at the same rate because they are both influenced by
the acceleration due to gravity. And because they fall at the same rate and the horizontal position has no influence this means since they fall the same distance they must take the same amount of time.
Now it might help to understand why the projected penny
falls further than the dropped penny in the same amount of time if you think of it like this. At any instance of time, the projected penny has a downward velocity component like this due to gravity. Now the dropped penny also has the same velocity
vector at the same instance of time; however, the projected penny also has a horizontal velocity component due to the flip I gave it initially. The overall velocity vector of the projected penny is therefore diagonal to the vertical component and has a magnitude that
is greater either of the components, so the average velocity of the projected penny is greater than that of the dropped penny. With the greater average velocity, the projected penny will obviously travel a greater amount of distance in the same amount of time
than the dropped penny will.