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11.3: Law of Addition of Velocities - Newtonian Mechanics

Last updated

Jul 20, 2022
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- 11.2: Galilean Coordinate Transformations
- 11.4: Worked Examples

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Suppose the object in Figure 11.1 is moving; then observers in different reference frames will measure different velocities. Denote the velocity of the object in frame $S$ by $\overrightarrow{\mathbf{v}}=d \overrightarrow{\mathbf{r}} / d t$ , and the velocity of the object in frame S′ by $\overrightarrow{\mathbf{v}}^{\prime}=d \overrightarrow{\mathbf{r}}^{\prime} / d t^{\prime}$ . Since the derivative of the position is velocity, the velocities of the object in two different reference frames are related according to

$\frac{d \overrightarrow{\mathbf{r}}^{\prime}}{d t^{\prime}}=\frac{d \overrightarrow{\mathbf{r}}}{d t}-\frac{d \overrightarrow{\mathbf{R}}}{d t} \nonumber$

$\overrightarrow{\mathbf{v}}^{\prime}=\overrightarrow{\mathbf{v}}-\overrightarrow{\mathbf{V}} \nonumber$

This is called the Law of Addition of Velocities.

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