Time and Mean Velocity
Date: 01/16/2003 at 23:03:21 From: Peter Subject: Why is "s = s0 + v0 t + 1/2 a t^2" true? I read books about how they come out with the formula and it's convincing but when I try to trace it manually, the answer is different. For example, init position s0 = 0 init velocity v0 = 0 acceleration a = 1 time t = 5 s = 0 + 0 + 1/2 (5^2) = 12.5 BUT ... 'v' is change of position in time and 'a' is change of velocity in time. So ... At t0, v0 = 0, then s0 = 0 At t1, v1 = 1 (i.e. v1 = v0 + a), then s1 = 1 (i.e. s1 = s0 + v1) Also, for other t's ... T S V --- --- --- 0 0 0 1 1 1 2 2 3 3 3 6 4 4 10 5 5 15 so at t = 5, the position 's' should be 15 and not 12.5 What am I understanding wrong on this? Thanks for your time.
Date: 01/17/2003 at 11:04:19 From: Doctor Rick Subject: Re: Why is "s = s0 + v0 t + 1/2 a t^2" true? Hi, Peter. Did you switch the S and V column labels? The second column correctly represents the velocity at time T seconds under constant acceleration of 1 meter/second per second. If I understand you correctly, you built the third column as follows: Take the position S at time T and add 1 second times the velocity at time T+1 to get the position at time T+1. This would be the correct thing to do IF the velocity of, say, 3 meters/second at time T=3 held constant for the entire one-second interval from T=2 to T=3. This is not the case. For instance, in the first interval (T=0 to 1) the velocity does not instantaneously jump from 0 at T=0 to 1 at T=0.0000000001 second and stay constant for a second. It gradually increases; it is 0.25 m/s at T=0.25, 0.5 m/s at T=0.5, etc. The MEAN velocity during this interval is (0+1)/2 = 0.5 m/s. If you use the mean velocity instead of the final velocity for each interval, your table will look like this: T V Vmean S 0 0 0 1 1 0.5 0.5 2 2 1.5 2.0 3 3 2.5 4.5 4 4 3.5 8 5 5 4.5 12.5 Well, what do you know! It comes out correctly this time. Does this help answer your question? To give a thorough answer I'd need to get into calculus, but this may be enough to show you that it makes sense. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Date: 01/18/2003 at 04:45:23 From: Peter Subject: Thank you (Why is "s = s0 + v0 t + 1/2 a t^2" true?) Thanks Doctor Rick for clearing up my confusion. Peter
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