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Finding time it will take an object to move from point a to point b

Posted 11 years ago
An object is observed to move along the x-axis with a varying velocity; v(x)=C[x(d?x)]^1/2, where C=9.0/s and d=48.0m
A. Use computer simulation to calculate the time it will take for the object to travel from x=0 to x=d.
Hint: Subdivide the x-axis into N parts of width ?x=d/N. The time for the object to move from xn to xn+1 is ?t=?x/v(x_a); here x_a=0.5(x_n + x_n+1). Now add up all the time intervals.
B. Calculate the max speed during this time (plot the function)

 c = 9.0;
 d = 48.0;
 n = 10;
 v[x_] := c (x (d - x))^(1/2)
 r = Range[0, d, d/n]
 p = Partition[r, 2, 1]
 xa = Mean /@ p
 deltaV = v /@ xa
 deltaT = (d/n)/deltaV
Plot[v[x], {x, 0, d}]
FindMaximum[v[x], x]
Here's a code from Mathematica program
Top speed is = 216 at x = 24
But how to find time it will take for x to travel from 0 to 48?
POSTED BY: Alan Suum
I think you shoud integrate 1/V over x from 0 to d to get time:
Clear[c, d, V]
V[x_] := c Sqrt[x (d - x)]
Integrate[1/V[x], {x, 0, d}, Assumptions -> d > 0]

You'll obtain pi/c and if c=9 then time is equal to 0.349066. Note that if you inrease the value of the parameter n up to, say, 10^7, then you get 0.349023. I'm not sure if the integration procedire is a possibility for your excercise but it's a good example of what it is useful for.
POSTED BY: Gregory Fridman
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