Consider the situation as shown. If a vehicle moves towards right with constant velocity `v_(0)`, find the velocity and acceleration of the block in terms of h and x, where x is the distance traveled by the vehicle towards right.

Explanation

As the total length of the string is constant,
`L=2h=(h-y)+(h^(2)+x^(2))^(1//2)`
`(dL)/(dt)=-(dy)/(dt)+(1)/(2)(h^(2)+x^(2))^(-1//2)(0+2x(dx)/(dt))`
`0=-v+(x)/(sqrt(h^(2)+x^(2)))v_(0)`
`v=(xv_(0))/((h^(2)+x^(2))^(1/2))`
Differentiating w.r.t. time t, we have
`(dv)/(dt)=(v_(0)[(h^(2)+x^(2))^(1//2)(dx)/(dt)-x xx(1)/(2)(h^(2)+x^(2))^(-1//2)xx2x(dx)/(dt)])/((d^(2)+x^(2)))`
`a=(v_(0)^(2)[(h^(2)+x^(2))^(3//2))`
`=(h^(2)v_(0)^(2))/((h^(2)+x^(2))^(3//2))`
OR
`v=v_(0)costheta=(v_(0)x)/(sqrt(h^(2)+x^(2)))`
If change in the length of spring is small, the force exerted by the spring is proportional to the change in length. the direction of the spring force is opposite to the change in length
`Fprop x`
`F=kx` (Hooke's law)
Where k is a spring constant or force constant unit of k is N/m.
For a spring kl=constant, if a spring of force constant k is divided in two parts, force constant of each part will be 2k.