Answer:
The correct answer will be "
". The further explanation is given below.
Explanation:
The potential energy will be,
⇒ ![U(x)= -\frac{C_{6}}{x^6}](https://tex.z-dn.net/?f=U%28x%29%3D%20-%5Cfrac%7BC_%7B6%7D%7D%7Bx%5E6%7D)
The expression of force will be,
⇒ ![F=-\frac{dU(x)}{dx}](https://tex.z-dn.net/?f=F%3D-%5Cfrac%7BdU%28x%29%7D%7Bdx%7D)
⇒ ![=-(C_{6}(-6)x^{-7})](https://tex.z-dn.net/?f=%3D-%28C_%7B6%7D%28-6%29x%5E%7B-7%7D%29)
⇒ ![=-\frac{6C_{6}}{x^{7}}](https://tex.z-dn.net/?f=%3D-%5Cfrac%7B6C_%7B6%7D%7D%7Bx%5E%7B7%7D%7D)
Force seems to be appealing because the expression has been negative. It therefore means that the force or substance is acting laterally in on itself.
The lemon trees will always be the same so they are the constants. The independent variable is the water amount because it is changing. The dependent variable is the amount of lemons because it is the variable being tested. (If we change the amount of water how many lemons will be produced?)
To solve the two parts of this problem, we will begin by considering the expressions given for gravitational potential energy and finally kinetic energy (to find velocity). From the potential energy we will obtain its derivative that is equivalent to the Force of gravitational attraction. We will start considering that all the points on the ring are same distance:
![r = \sqrt{x^2+R^2}](https://tex.z-dn.net/?f=r%20%3D%20%5Csqrt%7Bx%5E2%2BR%5E2%7D)
Then the potential energy is
![U = \frac{-GMm}{\sqrt{x^2+R^2}}](https://tex.z-dn.net/?f=U%20%3D%20%5Cfrac%7B-GMm%7D%7B%5Csqrt%7Bx%5E2%2BR%5E2%7D%7D)
PART A) The force is excepted to be along x-axis.
Therefore we take a derivative of U with respect to x.
![F = -\frac{dU}{dx}](https://tex.z-dn.net/?f=F%20%3D%20-%5Cfrac%7BdU%7D%7Bdx%7D)
![F = -\frac{d}{dx}(GMm(\frac{1}{R}-\frac{1}{\sqrt{x^2+R^2}}))](https://tex.z-dn.net/?f=F%20%3D%20-%5Cfrac%7Bd%7D%7Bdx%7D%28GMm%28%5Cfrac%7B1%7D%7BR%7D-%5Cfrac%7B1%7D%7B%5Csqrt%7Bx%5E2%2BR%5E2%7D%7D%29%29)
![F = \frac{GMmx}{(x^2+R^2)^{3/2}}](https://tex.z-dn.net/?f=F%20%3D%20%5Cfrac%7BGMmx%7D%7B%28x%5E2%2BR%5E2%29%5E%7B3%2F2%7D%7D)
This expression is the resultant magnitude of the Force F.
PART B) The magnitude of loss in potential energy as the particle falls to the center
![U = GMm(\frac{1}{R}-\frac{1}{\sqrt{x^2+R^2}})](https://tex.z-dn.net/?f=U%20%3D%20GMm%28%5Cfrac%7B1%7D%7BR%7D-%5Cfrac%7B1%7D%7B%5Csqrt%7Bx%5E2%2BR%5E2%7D%7D%29)
According to conservation of energy,
![\frac{1}{2}mv^2 = GMm (\frac{1}{R}-\frac{1}{\sqrt{x^2+R^2}})](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7Dmv%5E2%20%3D%20GMm%20%28%5Cfrac%7B1%7D%7BR%7D-%5Cfrac%7B1%7D%7B%5Csqrt%7Bx%5E2%2BR%5E2%7D%7D%29)
![\therefore v = \sqrt{2GM(\frac{1}{R}-\frac{1}{x^2+R^2})}](https://tex.z-dn.net/?f=%5Ctherefore%20v%20%3D%20%5Csqrt%7B2GM%28%5Cfrac%7B1%7D%7BR%7D-%5Cfrac%7B1%7D%7Bx%5E2%2BR%5E2%7D%29%7D)
Answer:
The horizontal distance traveled by the projectile is 60 m
Explanation:
Given;
initial horizontal velocity of the projectile, Vₓ = 30 m/s
time of the motion of the projectile, t = 2 s
The horizontal distance traveled by the projectile is given by the range of the projection;
X = Vₓt
X = 30 x 2
X = 60 m
Therefore, the horizontal distance traveled by the projectile is 60 m
Therefoe