- #1
Phys_Boi
- 49
- 0
BvU said:In 1 dimension ?
BvU said:And is $$F = -\displaystyle {GMm\over x^2} $$ or is $$F = +\displaystyle{GMm\over x^2} $$ as on the whyteboard ?
BvU said:Pity: https://www.wolframalpha.com/input/?i=x''+=+1/x^2
(The minus sign gave an analytical solution https://www.wolframalpha.com/input/?i=x''+=+-1/x^2 showing that it isn't an easy differential equation )
BvU said:I didn't do anything except enter the thing in wolframalpha !
BvU said:In the form you write it, it is the equation of motion for a repulsive gravitational force.
Perhaps you did mean a minus sign ?
https://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation#Modern_form
Chestermiller said:$$\frac{dv}{dt}=-\frac{GM}{x^2}$$If you multiply both sides of this equation by v=dx/dt, you get:$$v\frac{dv}{dt}=-\frac{MG}{x^2}\frac{dx}{dt}$$Both sides of this equation are exact differentials with respect to time.
Yes.Phys_Boi said:So is the following correct?
$$v dv = \frac{-MG}{x^2} dx$$
So how do you integrate over a time interval? That is to say, how do you find the velocity over the interval [0, t]?Chestermiller said:Yes.
Do you know how to solve for v as a function of x?Phys_Boi said:So how do you integrate over a time interval? That is to say, how do you find the velocity over the interval [0, t]?
Acceleration as a function of x to a function of time is a mathematical representation of how an object's acceleration changes as it moves along the x-axis over a period of time. It takes into account both the object's position and the rate at which its position changes over time.
Acceleration as a function of x to a function of time can be calculated using the formula a(x,t) = d^2x/dt^2, where dx/dt represents the object's velocity and d^2x/dt^2 represents its acceleration.
Acceleration as a function of x to a function of time is important in understanding the motion of objects in the real world. It allows scientists to analyze and predict how an object's acceleration changes as it moves along a specific path, providing valuable insights into the laws of motion.
Acceleration as a function of x to a function of time is closely related to Newton's laws of motion, specifically the second law which states that the net force acting on an object is equal to its mass times its acceleration. By analyzing the change in acceleration over time, scientists can determine the net force acting on an object and apply it to Newton's second law.
Yes, acceleration as a function of x to a function of time can be graphically represented on an x-t graph. The slope of the graph at any given point represents the acceleration of the object at that point in time. Additionally, the area under the graph represents the change in velocity over time, which can also be used to calculate acceleration.