Respuesta :
Answer:
The function F(x) for 0 < x < 5, the block's initial velocity, and the value of F(f).
(C) is correct option.
Explanation:
Given that,
Mass of block = 1.0 kg
Dependent force = F(x)
Frictional force = F(f)
Suppose, the following information would students need to test the hypothesis,
(A) The function F(x) for 0 < x < 5 and the value of F(f).
(B) The function a(t) for the time interval of travel and the value of F(f).
(C) The function F(x) for 0 < x < 5, the block's initial velocity, and the value of F(f).
(D) The function a(t) for the time interval of travel, the time it takes the block to move 5 m, and the value of F(f).
(E) The block's initial velocity, the time it takes the block to move 5 m, and the value of F(f).
We know that,
The work done by a force is given by,
[tex]W=\int_{x_{0}}^{x_{f}}{F(x)\ dx}[/tex].....(I)
Where, [tex]F(x)[/tex] = net force
We know, the net force is the sum of forces.
So, [tex]\sum{F}=ma[/tex]
According to question,
We have two forces F(x) and F(f)
So, the sum of these forces are
[tex]F(x)+(-F(f))=ma[/tex]
Here, frictional force is negative because F(f) acts against the F(x)
Now put the value in equation (I)
[tex]W=\int_{x_{0}}^{x_{f}}{(F(x)-F(f))dx}[/tex]
We need to find the value of [tex]\int_{x_{0}}^{x_{f}}{(F(x)-F(f))dx}[/tex]
Using newton's second law
[tex]\int_{x_{0}}^{x_{f}}{(F(x)-F(f))dx}=\int_{x_{0}}^{x_{f}}{ma\ dx}[/tex]...(II)
We know that,
Acceleration is rate of change of velocity.
[tex]a=\dfrac{dv}{dt}[/tex]
Put the value of a in equation (II)
[tex]\int_{x_{0}}^{x_{f}}{(F(x)-F(f))dx}=\int_{x_{0}}^{x_{f}}{m\dfrac{dv}{dt}dx}[/tex]
[tex]\int_{x_{0}}^{x_{f}}{(F(x)-F(f))dx}=\int_{v_{0}}^{v_{f}}{mv\ dv}[/tex]
[tex]\int_{x_{0}}^{x_{f}}{(F(x)-F(f))dx}=\dfrac{mv_{f}^2}{2}+\dfrac{mv_{0}^2}{2}[/tex]
Now, the work done by the net force on the block is,
[tex]W=\dfrac{mv_{f}^2}{2}+\dfrac{mv_{0}^2}{2}[/tex]
The work done by the net force on the block is equal to the change in kinetic energy of the block.
Hence, The function F(x) for 0 < x < 5, the block's initial velocity, and the value of F(f).
(C) is correct option.
