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Would it be a x ds = v x dv or a • ds = v • dv for describing curvilinear motion?

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- Thread starter Iqminiclip
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In summary: This is because in curvilinear motion, the acceleration is not necessarily perpendicular to the velocity, so the dot product takes into account the component of acceleration in the direction of displacement. In summary, the relation ads = vdv is not applicable in 3D curvilinear motion. Instead, the dot product equality a • ds = v • dv must be used to accurately describe the relationship between acceleration, displacement, and velocity. This is because in curvilinear motion, the acceleration may not be perpendicular to the velocity, so the dot product accounts for this.

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Would it be a x ds = v x dv or a • ds = v • dv for describing curvilinear motion?

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Hi sir, I was wondering why ds would be zero? Wouldn't a particle in circular motion have some sort of change in displacement over time? (e.g at 90° or so)andrewkirk said:

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Ah, I was interpreting yourIqminiclip said:Hi sir, I was wondering why ds would be zero? Wouldn't a particle in circular motion have some sort of change in displacement over time? (e.g at 90° or so)

In that case the equation has to be a dot product, not a multiplication, as one cannot multiply vectors. We have an equation for each of the three dimensions:

\begin{align}

a_1\,ds_1 &= v_1\,dv_1\\

a_2\,ds_2 &= v_2\,dv_2\\

a_3\,ds_3 &= v_3\,dv_3

\end{align}

Adding them together, we can represent this as:

$$\vec a\cdot \vec {ds} = \vec v\cdot \vec {dv}$$

This equation is known as the Fundamental Theorem of Calculus, and it represents the relationship between the rate of change of a variable (a or v) and the total change of another variable (ds or dv).

This equation is used in many scientific fields, such as physics, engineering, and biology, to calculate important quantities like velocity, acceleration, and displacement.

In this equation, a represents acceleration, ds represents displacement, v represents velocity, and dv represents change in velocity.

No, this equation can be used for both one-dimensional and multi-dimensional motion. In multi-dimensional motion, the variables would represent the components in each direction (e.g. x, y, and z).

Yes, this equation can be derived from the chain rule of calculus, which states that the derivative of a composite function is equal to the product of the derivatives of its individual components.

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