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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 2αv3
The relation between time and the distance is given by
t=αx2+βx
Differentiate 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 get
0=2α(xa+v2)+βa
βa=2α(xa+v2)=2αxa2αv2
βa+2αxa=2αv2
a=2αv2β+2αx. . . . . . . . .(5)
Substitute equation (3) and (4) in equation (1), we get
1=2xαv+βv
2xαv=1βv
2αx=1βvv. . . . . . . . .(6)
Substitute equation (6) in equation (5), we get
a=2αv2β+1βvv=2αv2βv+1βv×v
a=2αv3
The retardation is 2αv3.
The correct option is A.

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