velocity question The velocity of an object as a function of time is shown in fi

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velocity question

The velocity of an object as a function of time is shown in fig. P9.3 the acceleration is constant during the first four seconds of motion, so the velocity is a linear function of time with v(t) = 0 at t = 0 and v(t) = 100 ft/s at t = 4 s. The velocity is constant during the last 6 s. Estimate the total distance covered: i.e., estimate the area, A, under the velocity curve using five rectangles of equal width (delta t = 10/5 = 2s). Now, estimate the total distance covered using 10 rectangles of equal width. Calculate the exact area under the velocity curve, i.e., find the total distance traveled by evaluating the definite integral delta x = Calculate the exact area by adding the area of the triangle and the area of the rectangle firmed from the velocity curve. A particle is accelerated along a curved path of length l under the action of an applied force

Explanation / Answer

we have got the velocity fucnction as v=25 t for th1st 4 second a> taking interval of 2 s at 2 s velocity=2*25=50ft/s thus area of trapezium=0.5*(0+50)*2 similarly the trapezium formed by next 2 s will have area 0.5*(50+100)*2 thus total distance travelled=50+150+200+200+200=800ft b>if we take the time interval 1 s then again trapezium and rectangle will be formed aplying formulae we get distance covered=0.5*(0+25)*1+0.5*(25+50)*1+0.5*(50+75)*1+0.5*(75+100)*1+100+100+100+100+100+100 =712.5 ft c>exact are is formed by integration so integrating dx/dt=25t for the 1st 4 s and v=100 for the next 6 second we get 200+600= 800ft d>area of triangle=0.5*4*100=200 and area of rectangle=6*100 =600 ft thus total distance=800ft

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