Answer :
Answer:
7.136 m/s
Explanation:
We are given that
Mass =m=7.41 kg
[tex]F(x)=Ax^3 hat{x}[/tex]
Where [tex]A=-4 N/m^3[/tex]
[tex]x_i=2.05 m[/tex]
[tex]u=v_i=6.51 m/s[/tex]
We have to find the speed of body when it is position [tex]x_f=-2.65 m[/tex]
According to work energy theorem
[tex[K_f-K_i=Work\; done[/tex]
[tex]\frac{1}{2}mv^2-\frac{1}{2}mu^2=\int_{2.05}^{-2.65}-4x^3 dx[/tex]
[tex]\frac{1}{2}m(v^2-u^2)=-[x^4]^{2.05}_{-2.65}[/tex]
[tex]\frac{1}{2}(7.41)(v^2-(6.51)^2)=-((2.05)^4-(2.65)^2)[/tex]
[tex]\frac{1}{2}(7.41)(v^2-42.3801)=31.6545[/tex]
[tex]v^2-42.3801=\frac{31.6545\times 2}{7.41}[/tex]
[tex]v^2-42.3801=8.5437[/tex]
[tex]v^2=8.5437+42.3801=[/tex]
[tex]v=\sqrt{8.5437+42.3801}[/tex]
[tex]v=7.136 m/s[/tex]
The speed of the body at its final position is of 7.136 m/s.
What is Work - Energy Theorem?
As per the work - energy theorem, "When some amount of force is applied on an object, then the work done on the object is equal to the change in kinetic energy of object."
Given data:
The mass of body is, m = 7.41 kg.
The conservative force is, F(x) = Ax³. (here A is force constant)
The value of force constant is, A = - 4 Nm³.
The initial position of body is, xi = 2.05 m.
The initial speed of body is, vi = u = 6.51 m/s.
The final position of body is, xf = - 2.65 m.
The given problem is based on the concept and fundamentals of work - energy theorem.
As per the work energy theorem, the change in kinetic energy equals the work done.
Then,
[tex]\Delta KE = W\\\\\dfrac{1}{2}mv^{2}-\dfrac{1}{2}mu^{2} = \int\limits^{-2.65}_{2.05} {F(x)} \, dx[/tex]
here, v is the final speed of the body.
Solve by substituting the values as,
[tex]\dfrac{1}{2}mv^{2}-\dfrac{1}{2}mu^{2} = \int\limits^{2.05}_{- 2.65} {Ax^{3}} \, dx\\\\\\\dfrac{1}{2}m(v^{2}-u^{2}) = \int\limits^{2.05}_{-2.65} {-4x^{3}} \, dx\\\\\\\dfrac{1}{2} \times 7.41 \times (v^{2}-(6.51)^{2}) = -[(2.05)^{4}-(-2.65)^{4}]\\\\\\v=\sqrt{8.5437 + 42.3801}\\\\v = 7.136\;\rm m/s[/tex]
Thus, we can conclude that the speed of the body at its final position is of 7.136 m/s.
Learn more about the speed and distance here:
https://brainly.com/question/12759408