Question

Displacement(x) of a moving body changes with time(t) as x^2=t^2+2t+3 . Then find the relation between displacement(x) and acceleration(a) of this body?

Answer 1

Given relation between displacement `x` and time `t`
`x^2=t^2+2t+3`
acceleration `a=(d^2x)/dt^2`

First differential
`d/dtx^2=d/dt(t^2+2t+3)`
`2x(dx)/dt=(2t+2)`
`x(dx)/dt=t+1`

Differentiating again
`d/dt(x(dx)/dt)=d/dt(t+1)`
`x xxa+(dx)/dt(dx)/dt=1`
`=>a=(1-((dx)/dt)^2)/x`
can also be written as
`=>a=(1-(dotx)^2)/x`

Is the required expression.