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A particle moving along x-axis has accel...

A particle moving along x-axis has acceleration `f`, at time `t`, given by `f = f_0 (1 - (t)/(T))`, where `f_0` and `T` are constant.
The particle at `t = 0` has zero velocity. In the time interval between `t = 0` and the instant when `f = 0`, the particle's velocity `(v_x)` is :

A

1) `f_(0)T`

B

2) `frac(1)(2)f_(0)T^(2)`

C

3) `f_(0)T^(2)`

D

4) `frac(1)(2)f_(0)T`

Text Solution

Verified by Experts

The correct Answer is:
D
Given the acceleration of the particle
`f=f_(0)(1-frac(t)(T))`
or `f=frac(dv)(dt)=f_(0)(1-frac(t)(T))`
or `dv=f_(0)(1-frac(t)(T))dt`
Integrating above equation,
`int_(0)^(v)dv= int_(0)^(1)(1-(1)/(T))dt`
`v= [f_(0)t-(f_(0))/(T).(t)^(2)/(2)`
If f=0
Then `f= f_(0)(1-(t)/(T))= 0`
`1-(t)/(T)=0 `or t=T
Subsitutuing = tmEq(ii) then velocity
`V_(x)T-(f_(0)/(t).(T^(2))/(2)= f_(0)-f_(0)T)/(2)=(1)/(2)f_(0)T`
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