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The Physics of Spiderman's Swing

Can Spiderman land at the blue mark?
In the movies and in comic books, when Spiderman swings from building to building on his web, he releases at the top of his swing and travels in the horizontal direction to land on the side of a building. However, releasing at the top of his swing means he would have no velocity in the horizontal direction and therefore he would drop straight down. His greatest velocity in the horizontal direction would come at the bottom of his swing. But if the bottom of his swing is close to the ground, he may not travel far enough before hitting the ground. The maximum distance he would travel in the horizontal direction before hitting the ground comes at some point in the upward arc of his swing.
In this Demonstration, parabolic paths show how he would travel based on the release point in his swing. You can move the wall of a nearby building closer or farther away from the swing, and you can select a point on the wall as the target point for Spiderman to land. You can then adjust the release point to find the optimum path to reach the target. For most target points, there are two possible release points that achieve the target. However, there is only one release point that would reach the highest possible target point on the wall at any given distance.
Move the slider to adjust the distance to the wall of a nearby building. Then move the slider to select a point on the wall as the target for Spiderman to reach. Finally, move the slider to select the release point in the swing to try to reach the target point, and check the "release" box. Also try to reach the target point on the wall with another release point if possible.

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DETAILS

The motion of the pendulum of length can be determined by using the principle of conservation of mechanical energy
,
which gives
.
From this,
.
The solution of this differential equation gives the time it takes the pendulum to reach a position . Using the inverse of this function, , we find the position of the pendulum at time as
,
.
When the bob is released at time at position , the projectile motion of the bob is given by the equations
,
,
where the velocity .
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