Maximum Speed and Acceleration of Bird-please help!

Question

The motion of a nightingale's wingtips can be modeled as simple harmonic motion. In one study, the tips of a bird's wings were found to move up and down with an amplitude of 8.0 cm and a period of 0.82 s.

1. What is the wingtips' maximum speed?
2. What is the wingtips' maximum acceleration?

Answer 1

Hi Lauren M.,

So, write your equation for displacement (where the wingtip is on a vertical axis) in terms of a sin function. That would be D = 8* sin ((2*pi/0.82)*t) -- because amplitude is 1/2 the distance peak to trough, and every t=0.82 s, you have done 2*pi movement "along" the sin wave (graphed as t as x-axis).

You should know by now that the velocity V (well, here, speed, don't think your instructor cares up vs. down!) is the first derivative of D with respect to t, and the acceleration is the 2nd derivative of D with respect to t (in other words, the first derivative of V with respect to time). And you know how to calculate those?.... So do that first.

But, we aren't done with this problem by any means! The vertical displacement is only a portion of the total movement IN SPACE because you never see swallows hovering! So you could look up some data on birds' air speeds, and assuming NON-AFRICAN swallows (see Monty Python for background) add the constant (lateral) air speed to the wingtip displacement equation. (It's a Pythagorean triangle calculation). Then take your derivatives. If you don't know how to apply the Chain Rule in that, just set it up, I'm sure your instructor will be impressed.

Bear in mind, the swallows do all that, and catch their dinner in the process --

-- Cheers, -- Mr. d.