Let’s ignore air resistance for a second. If you point a cannon, aim an arrow or throw a basketball, the trajectory that gets you furthest will be at 45 degrees to the vertical. So the same must be true for Tarzan on a rope swing. He ought to let go when the rope is at 45 degrees to the vertical, right?
Not so, says Hiroyuki Shima at the University of Yamanashi in Japan, who today takes us through some straightforward calculations to show the answer is not quite as intuitive as you might imagine.
Shima begins by defining the question as in the diagram above. The problem, of course, is that Tarzan’s horizontal velocity reaches a maximum when the rope is at the bottom of its swing, at 0 degrees to the vertical.
By hanging on beyond this point, Tarzan begins to convert some of this horizontal velocity into vertical speed, which sends him on an upwards parabolic trajectory that can increase his time in the air and therefore the distance he travels along the ground .
The balance that has to be struck is between the lost horizontal velocity and the vertical velocity gained. When does this maximise the horizontal distance he travels?
Shima shows first that to maximise the distance, the angle of the rope at the point of release should always be less than 45 degrees. That’s in stark contrast to the case of throwing or firing a missile, which is why this problem is a little counterintuitive.
He goes on to show that Tarzan cannot significantly increase his flight duration by hanging on to the rope much beyond the lowest point of the swing. “The ﬂight duration is not signiﬁcantly altered by acquiring the upward component,” he says.
So a small angle of release is ideal, although not too small an angle.
In fact there is no simple rule for maximising the horizontal flight distance. It turns out this depends on a number of factors, such as rope swing’s distance off the ground, the length of the rope and the angle of the rope when Tarzan begins his swing as well as the angle of the rope at the point of release.
So there you have it: a well posed problem with some interesting physics to boot. Johnny Weissmuller would be pleased.
Or as he would put it: Aaaaaaaaaayaaahh-eeeeeeeeeeeeyaaaaaaah-aaaaaaaaaaaaaaaaahaaah.
Ref: arxiv.org/abs/1208.4355: How Far Can Tarzan Jump?