Horizontal displacement is calculated similarly to vertical displacement. First you need to know the time for the jump from take-off to landing. You use the time for the whole jump because you are calculating the distance of the whole jump. In situations where the object undergoing projectile motion takes-off and lands on the same level, the total time is just twice the time-up.
The logic behind this fact is that the object travels the same distance up and down. Since gravity is constant, gravity will decrease the vertical velocity of the object on the way up the exact same amount it will increases the vertical velocity of the object on the way down. Thus the vertical velocity at take-off is identical in magnitude, but opposite in direction, to the vertical velocity at landing. The following two figures, one of just a vertical jump and one of a figure skating jump illustrate this concept.
Since the object undergoes a similar change in velocity on the way up as on the way down, and the up and down distances are the same, the time must also be the same.
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If an object takes-off and
lands on surfaces of identical height, then the time-up equals the time
down, and the vertical take-off velocity equals minus the vertical landing
velocity.
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In figure skating, the skaters take-off and land on a level surface. However, remember that we are measuring distances from the center of mass, not the surface of the ice. Thus, only if the center of mass of the skater is at the same height at take-off and landing will the time up and time down be equal. In some jumps with some skaters, the center of mass will be at the same level at take-off and landing, in others it will not. Remember that the center of mass depends on the position of the skater at these two instances. The following two jumps show one skater who has a similar center of mass positions at take-off and landing and one who has very different center of mass positions at take-off and landing position.
When substituting values for these variables, be sure they match with what you are calculating. When calculating horizontal displacement:
Continuing the example form before:
The take-off velocity and angle for the jump are provided below. Try calculating jump distance. Then, continue on through the illustration to see the jump and the solution.
Instead of us providing take-off velocities of a jump for you, why don't you pick some take-off velocities and angles and see how far the skater would jump and what the trajectory of his jump would look like.
You can pick take-off speeds and angles, see the calculated vertical and horizontal take-off velocities, jump heights, and jump distances.
Realistic numbers for a skater would be a take-off velocity between 4 to 6 m/s and30 to 45 degree take-off angles, but you can try any values.
Go to the Projectile Calculator
The only change you need to make is when calculating total time. In this case, you can not just double the time-up. Instead, calculate time-up as usual (also calculate jump height while you're at it -- it will come in handy), then calculate time-down. Time-down can actually be calculated using the displacement formula. All you need to know is the difference in height between the take-off and landing positions.
For example, in the following figure, the skater takes-off at a 43 degree angle at 10 m/s, and lands with her center of mass 10 cm lower than it was at take-off.
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