# Deriviate proof d/dt[r (v a)]= r(v a)

• agentnerdo
In summary, The question asks to prove the equality of two sides of the equation \frac{d}{dt}[r (v x a)] = r (v x a*) where a* is the derivative of a. The solution involves applying the chain rule and using the properties of cross products and dot products.
agentnerdo
Hey guys, I really do not even know how to get this question started.

## Homework Statement

\frac{d}{dt}[r (v x a)] = r (v x a)

the last a is supposed to have a period on top

as such it is \frac{d}{dt}[r (v x a)] = r (v x \frac{d}{dt}a)

d, v, and a are position, velocity, and acceleration

the last a, after the = sign is d/dt of a, as mentioned above

## The Attempt at a Solution

I do not even know how to start...I tried doing different degrees of deriviatives of both left and right sides but, it seems like I need to get ride of one unit of /s (time).

that was an epic fail on trying to use brackets...

derivative of [r (v times a)] = r ( v times a*)

a* is the derivative of a.

question asks to prove that both sides are equal.

Simply apply the chain rule and remember that the cross product of a vector with itself is 0, and that $$\vec a \cdot (\vec a \times \vec b)=0$$ (Convince yourself this is true, because a x b is perpendicular to a, and the dot product of a vector with another vector to which it is orthogonal, is 0)

## 1. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function with respect to its independent variable. In other words, it shows how much a function is changing at a particular point.

## 2. What does the notation d/dt represent in the equation d/dt[r (v a)]= r(v a)?

The notation d/dt represents differentiation with respect to time. It is commonly used in calculus to indicate that the derivative of a function is being taken with respect to the independent variable, in this case, time.

## 3. What is the proof for the derivative d/dt[r (v a)]= r(v a)?

The proof for this derivative involves using the product rule of differentiation, which states that the derivative of a product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function. Applying this rule to the equation d/dt[r (v a)]= r(v a), we can obtain the derivative by taking the derivatives of each term separately and then combining them using the product rule.

## 4. How is the derivative d/dt[r (v a)]= r(v a) used in science?

This derivative is commonly used in physics and engineering to calculate the acceleration of an object in motion. By taking the derivative of the position function with respect to time, we can obtain the velocity function, and by taking the derivative of the velocity function with respect to time, we can obtain the acceleration function.

## 5. Can you provide an example of how to use the derivative d/dt[r (v a)]= r(v a)?

Sure, let's say we have a particle with position function r(t) = 3t^2 + 2t + 1. To find its acceleration at a specific time t, we can use the derivative d/dt[r (v a)]= r(v a). First, we find the velocity function by taking the derivative of r(t), which is v(t) = 6t + 2. Then, we take the derivative of v(t) to find the acceleration function, which is a(t) = 6. This means that the acceleration of the particle at any given time is always 6 units per second squared.

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