# Newton's second law dp/dt version?

• Steve Drake
In summary, the momentum version of Newtons second law states that the acceleration is the second order derivative of position.
Steve Drake
Hello,

I am confused by the momentum version of Newtons second law...

So we know
$$\bar{F}=m\bar{a}=m\left(\frac{d\hat{v}}{dt}\right)$$
and that
$$\bar{\rho}=m\bar{v}=m\left(\frac{d\bar{x}}{dt}\right)$$

so is

$$\frac{d\bar{p}}{dt}=m\frac{d\left(\frac{d\bar{x}}{dt}\right)}{dt}$$

What I mean is this bit $$\frac{d\left(\frac{d\bar{x}}{dt}\right)}{dt}$$ somehow equal to $$\bar{a}$$

Thanks

Yes, since acceleration is the (first order) time derivative of velocity and velocity is the (first order) derivative of position, the acceleration is said to be the second order derivative of position, which can be written as ##a = \frac{d^2x}{dt^2}##

You can read more about other notations for higher derivatives on [1]

[1] https://en.wikipedia.org/wiki/Derivative

Force F = dp/dt, this result summarizes Newton's first/second law, to prove this, we know that F = ma = m*dv/dt, mass is invariant so we can treat it as a constant, which yields to m*dv/dt = d(m*v)/dt = dp/dt, If no force is acting on an object F = 0 = dp/dt, p is constant from which it follows Newton's first law and momentum conservation, Good luck :p

Steve Drake said:
Hello,

I am confused by the momentum version of Newtons second law...

So we know
$$\bar{F}=m\bar{a}=m\left(\frac{d\hat{v}}{dt}\right)$$
and that
$$\bar{\rho}=m\bar{v}=m\left(\frac{d\bar{x}}{dt}\right)$$
Using the product rule:

$$\vec{F} = \frac{d\vec{p}}{dt}=\frac{d}{dt}(m\vec{v}) = m\frac{d\vec{v}}{dt} + \frac{dm}{dt}v$$

If m is constant, dm/dt = 0 so:

$$\vec{F} = m\frac{d\vec{v}}{dt} = ma = m\frac{d}{dt}v = m\frac{d}{dt}(\frac{d\vec{x}}{dt})$$

AM

Last edited:
Thanks guys, forgot about the chain rule for differentiation.

So in general, whenever there is a $$\frac{d^{2}}{dx}$$ then it can be thought of two separate derivatives, each giving their own result. But the $$^2$$ means it skips the first result and we go right to the second?

Steve Drake said:
Thanks guys, forgot about the chain rule for differentiation.

So in general, whenever there is a $$\frac{d^{2}}{dx}$$ then it can be thought of two separate derivatives, each giving their own result. But the $$^2$$ means it skips the first result and we go right to the second?
Actually, I misspoke. It is the product rule not the chain rule. I have corrected the error.

The ##\frac{d^{2}x}{dt^2}## signifies the second derivative with respect to time - the derivative with respect to time of (the derivative with respect to time of x).

AM

## 1. What is Newton's Second Law (dp/dt version)?

Newton's Second Law, also known as the Law of Acceleration, states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. In mathematical terms, this can be expressed as F = ma, where F is the force, m is the mass, and a is the acceleration.

## 2. How is the dp/dt version of Newton's Second Law different from the F=ma version?

The dp/dt version of Newton's Second Law is a slightly more advanced form of the equation, where the change in momentum (dp) over time (dt) is equal to the net force acting on an object. This is a more general form of the equation that can be used for objects with varying mass or forces acting on them.

## 3. What is the significance of dp/dt in Newton's Second Law?

The dp/dt portion of the equation represents the rate of change of an object's momentum, which is a measure of its motion. This term allows for a more accurate and comprehensive understanding of how forces and motion are related.

## 4. How does Newton's Second Law relate to real-world situations?

Newton's Second Law is a fundamental principle in physics and is used to explain the motion of objects in the real world. It helps us understand the relationship between forces, mass, and acceleration and can be applied to a wide range of situations, from the movement of planets to the motion of everyday objects.

## 5. What are some examples of Newton's Second Law in action?

Some common examples of Newton's Second Law in action include a car accelerating when the gas pedal is pressed, a ball being thrown and traveling through the air, and a rocket launching into space. In each of these cases, the force applied to the object causes it to accelerate, according to the principles of Newton's Second Law.

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