I checked the limiting cases (small h and big h) and it looks ok.Īddendum- I'd left out the expression for distance h in terms of time, t, thinking it would be even messier. Time= (V F/g)*(integral over of dx/sqrt(x^2-1) For the case where the friction is proportional to velocity squared, here's what I get: So Mathematica didn't download properly, and I had a chance to sleep on it. Why is the time interval now written as t If we know three of these five kinematic variables for an object under constant acceleration, we can use a kinematic formula, see below, to solve for one of the unknown variables. I might have to use a program, or look it up somewhere, if you want that formula too. The kinematic formulas are a set of formulas that relate the five kinematic variables listed below. The more common case, like for a ball, where the friction becomes proportional to the square of the velocity after a short period of acceleration is harder to solve. Then after time t the height fallen, h, is given by: From the third equation of motion, we have \(x(t) = 30t-5t^2\).Ok, I'll do the case of some very fluffy thing where the friction is proportional to the velocity, falling in gravitational acceleration from an initial velocity of zero, reaching terminal velocity V F.The stone hits the ground when \(t=6\).Hence, the maximum height reached by the stone is 45 metres. When the stone reaches the highest point, \(v=0\). We can find the maximum height using the fourth equation of motion, \(v^2 = u^2+2ax\).Thus the time taken to return to the ground is 6 seconds. To find the time taken to return to the ground, we substitute \(x=0\) to obtain If we assume that the rate of change of velocity (acceleration) is a constant, then the constant acceleration is given by After three seconds, the velocity is still decreasing, but the speed is increasing (the particle is going faster and faster). At three seconds, the particle is momentarily at rest. Over the first three seconds, the particle's speed is decreasing (the particle is slowing down). The velocity–time graph for this motion is shown below it is the graph of \(v(t)=6-2t\). First, the metric of time is independent of the metric of space. The first two concern the metrical properties of time, whereas the third is more a property of the dynamics than of time itself. In general, the velocity of the particle is \(6-2t\) m/s after \(t\) seconds. Time in classical physics does have a number of remarkable properties, of which three will be mentioned here. ![]()
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