The amplitude and maximum velocity will be respectively `X= 3 sin 2t + 4 cos 2t`The amplitude and maximum velocity will be respectively

Options

A. `5, 10`
B. `3, 2`
C. `4, 2`
D. `3, 4`

Answer

A

Explanation

`x=3sin2t+4cos2t`
`=5[3/5sin2t+4/5cos2t]`
`=5[cosalphasin2t+sinalphacos2t]`
`x=5sin(2t+alpha)`
`v=(dx)/(dt)=5xx2cos(2t+alpha)`
`=10cos(2t+alpha)`
Amplitude `A=5`, `v_(max)=10`
Alternatively:
`x=3sin2t+4cos2t`
`=3sin2t+4sin(2t+pi/2)`
`x=Asin(2t+alpha)`

`A=sqrt(3^2+4^2)=5`
`tanalpha=4/3`
`x=5sin(2t+alpha)`
`v=(dx)/(dt)=5xx2cos(2t+alpha)`
`=10cos(2t+alpha)`
v is maximum when `cos(2t+alpha)=1`
`v_(max)=10`