A block mass m (0.25 kg) is pressed against (but is not attached to) an ideal spring of force constant k (100 N/m) and negligible mass, compressing the spring a distance 0.1 m. After it is released, the box slides along a leveled surface for 2 m before fully stops. If we repeat this experiment but instead use a spring having force constant 2k,

a. the box will go up the incline twice as high as before.
b. just as it moves free off the spring, the kinetic energy of the box will be twice as great as before.
c. just as it moves free off the spring, the speed of the box will be
d. All the above choices are correct.
e. None of the above choices are correct.

d. All the above choices are correct.

Explanation:

When a spring of spring constant k is compressed by distance x , the potential energy stored in it is equal to

E = 1/2 k x²

If spring constant is 2 k , potential energy stored

E = 1/2 2k x²

= k x²

which is twice the earlier potential energy.

In the first case , the energy of spring is imparted to box . The energy given to box is spent by frictional force due to which box comes to rest.

So energy of box acquired from spring = work done by frictional force.

So energy of box acquired from spring =  F X d  , F is frictional force , d is displacement .

In the second case ,

energy acquired by box becomes  two times

Work done by frictional force will also become two times to put box at rest

So displacement will be two times ( because frictional force is constant )

so option a is correct .

option b is also correct .

Because kinetic energy  of box will be twice as explained above .

So option d will be correct.

we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .