Inactive Tutor answered • 03/02/21
Total distance traveled is given by the integral of speed, ie the absolute value of the velocity function, over the given time interval. This is relevant because the velocity is negative between t = 3 and t = 4. Thus displacement ≠ total distance. We could use a calculator and use the absolute value function to calculate ∫04 |6 - 2t| dt, or we can do it by hand by splitting up the integral into 2 separate ones as I will do here:
total dist = ∫03 (6-2t)dt + ∫34(2t-6)dt = (6t - t2) ]03 + (t2 - 6t)]34 = 9 - 8 - (-9) = 10
Btw we can calculate this distance easily using geometry, given that the linear velocity function forms two triangles with the x-axis, the first with height = 6 and base = 3 so A = 9, and the second with height = -2 and base = 1 so A = 1.
Inactive Tutor
Great question! The reason both terms in the integrand change signs on the interval from 3 to 4 is that the function 6 - 2t becomes negative at t = 3. The way that the absolute value function works algebraically is that the absolute value of an expression is = to itself whenever it is positive, and equal to its opposite when it is negative. So, for example |x - 3| = x - 3 for x > 3 but it = 3 - x when x < 3. Try some values for x in the above expression to convince yourself that that’s how it works.05/22/21
Gi G.
How come the 6 - 2t switch places in the integral from four to three? Shouldn't it stay consistent throughout?05/22/21