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Question

# The relation between time and distance is t = αx2+βx, where α and β are constants. The retardation is

A
2αv3
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B
2βv3
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C
2αβv3
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D
2β2v3
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Solution

## The correct option is B 2αv3The relation between time and the distance is given byt=αx2+βxDifferentiate the above equation with respect to time,1=2xαdxdt+βdxdt. . . . . . .(1)Differentiate with respect to time,0=2α(xdx2dt2+dxdtdxdt)+βdx2dt2. . . . .(2)Velocity, v=dxdt. . . . . . .(3)Acceleration, a=d2xdt2. . . . . . . .(4)Substitute equation (3) and (4) in equation (1), we get0=2α(xa+v2)+βaβa=−2α(xa+v2)=−2αxa−2αv2βa+2αxa=−2αv2a=−2αv2β+2αx. . . . . . . . .(5)Substitute equation (3) and (4) in equation (1), we get1=2xαv+βv2xαv=1−βv2αx=1−βvv. . . . . . . . .(6)Substitute equation (6) in equation (5), we geta=−2αv2β+1−βvv=−2αv2βv+1−βv×va=−2αv3The retardation is 2αv3.The correct option is A.

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