What Does -∂V/∂x Represent in Newton's Law?

• rb120134
In summary: Partial derivatives are important in physics because they let us calculate rates of change of functions with lots of variables.
rb120134
Homework Statement
What does -∂V/∂x mean?
Relevant Equations
F= ma
In one of my textbooks about quantum mechanics, they mention a vehicle moving in a straight line along the x axis. With Newtons first law they take the second derivative from a which is
d^2x/dt^2 and that should be equal to
-∂V/∂x. What exactly does -∂V indicate?
The complete equation becomes
F=ma md^2x/dt^2= -∂V/∂x. It is a partly derivative. What does " -∂V/∂x" indicate, which units belong to -∂V and
∂x? They were talking about potential energy in the textbook.

rb120134 said:
Homework Statement: What does -∂V/∂x mean?
Homework Equations: F= ma

In one of my textbooks about quantum mechanics, they mention a vehicle moving in a straight line along the x axis. With Newtons first law they take the second derivative from a which is
d^2x/dt^2 and that should be equal to
-∂V/∂x. What exactly does -∂V indicate?
It doesn't indicate anything! The "-∂V" alone means nothing without the accompanying "/∂x".

The complete equation becomes
F=ma md^2x/dt^2= -∂V/∂x. It is a partly derivative
"partial derivative", not "partly derivative"
What does " -∂V/∂x" indicate, which units belong to -∂V and
∂x? They were talking about potential energy in the textbook.
As I said above, "-∂V" and "∂x" have no meaning separately. "-∂V/∂x" is the "negative of the partial derivative with respect to x. You titled this "second derivative of Newton's Law" and mentioned "partial derivative". Can we assume that you know what "derivatives" and "partial derivatives" are? The derivative of a function is instantaneous rate of change of the function. If a function, like V, depends on more that one variable (here V is defined in "three space" so depends on the variables x, y, and z and might depend on time, t) then the partial derivative of V with respect to x is the instantaneous rate of change of V as x changes, all other variables held constant.

What is the second derivative of Newton's law?

The second derivative of Newton's law is the rate of change of acceleration, or the rate of change of the rate of change of velocity, with respect to time.

Why is the second derivative of Newton's law important?

The second derivative of Newton's law allows us to analyze the changes in acceleration over time, which can provide valuable insights into the movement and behavior of objects under the influence of forces.

How is the second derivative of Newton's law calculated?

The second derivative of Newton's law can be calculated by taking the derivative of the first derivative (acceleration) with respect to time. It can also be calculated by taking the second derivative of the position function with respect to time.

What is the physical interpretation of the second derivative of Newton's law?

The second derivative of Newton's law represents the rate of change of acceleration, which is a measure of how quickly the velocity is changing. It can also be thought of as the curvature of the position-time graph, providing information about the direction and magnitude of the acceleration.

In what situations is the second derivative of Newton's law most useful?

The second derivative of Newton's law is most useful in analyzing and understanding the behavior of objects under varying forces, such as in projectile motion, circular motion, and simple harmonic motion.

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