![]() ![]() ![]() The vestibular system comes equipped with sensors that detect angular acceleration (the semicircular canals) and sensors that detect linear acceleration (the otoliths). Part of this labyrinth is dedicated to our sense of hearing (the cochlea) and part to our sense of balance (the vestibular system). Located deep inside the ear, integrated into our skulls, lies a series of chambers called the labyrinth. The human body comes equipped with sensors to sense acceleration and jerk. Jerk is not just some wise ass physicists response to the question, "Oh yeah, so what do you call the third derivative of position?" Jerk is a meaningful quantity. The SI unit of jerk is the meter per second cubed. This makes jerk the first derivative of acceleration, the second derivative of velocity, and the third derivative of position. Jerk is the rate of change of acceleration with time. Let's apply it to a situation with an unusual name - constant jerk. The method shown above works even when acceleration isn't constant. Algebra works and sanity is worth saving. Not that there's anything wrong with that. ![]() We'd be back to using algebra just to save our sanity. If acceleration varied in any way, this method would be uncomfortably difficult. However, it really only worked because acceleration was constant - constant in time and constant in space. Here's what we get when acceleration is constant… dvĬertainly a clever solution, and it wasn't all that more difficult than the first two derivations. Get things that are similar together and integrate them. We get one derivative equal to acceleration ( dv dt) and another derivative equal to the inverse of velocity ( dt ds). We'll use a special version of 1 ( dt dt) and a special version of algebra (algebra with infinitesimals). By logical extension, it should come from a derivative that looks like this… dvīut what does this equal? Well nothing by definition, but like all quantities it does equal itself. The third equation of motion relates velocity to position. The second equation of motion relates position to time. We essentially derived it from this derivative… dv The first equation of motion relates velocity to time. We need to play a rather sophisticated trick. We can't just reverse engineer it from a definition. Unlike the first and second equations of motion, there is no obvious way to derive the third equation of motion (the one that relates velocity to position) using calculus. This gives us the position-time equation for constant acceleration, also known as the second equation of motion . Instead of differentiating position to find velocity, integrate velocity to find position. aĪgain by definition, velocity is the first derivative of position with respect to time. If we assume acceleration is constant, we get the so-called first equation of motion . This gives us the velocity-time equation. Instead of differentiating velocity to find acceleration, integrate acceleration to find velocity. Take the operation in that definition and reverse it. By definition, acceleration is the first derivative of velocity with respect to time. Discussion motion with constant accelerationĬalculus is an advanced math topic, but it makes deriving two of the three equations of motion much simpler. ![]()
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