Vector Transformations Under Sign Change
The question asks to identify which vector quantity does not change sign when the position vector $\overrightarrow{r} = x\hat{i} + y\hat{j} + z\hat{k}$ changes its sign to $-\overrightarrow{r}$. This transformation signifies a spatial inversion.
Analysis of Vector Quantities
We examine the behavior of each option under the condition $\overrightarrow{r} \to -\overrightarrow{r}$.
Velocity ($\overrightarrow{v}$)
Velocity is the time rate of change of position: $ \overrightarrow{v} = \frac{d\overrightarrow{r}}{dt} $ If $\overrightarrow{r}$ changes sign, velocity also changes sign: $ \overrightarrow{v'} = \frac{d(-\overrightarrow{r})}{dt} = -\frac{d\overrightarrow{r}}{dt} = -\overrightarrow{v} $ Velocity flips sign.
Acceleration ($\overrightarrow{a}$)
Acceleration is the time rate of change of velocity: $ \overrightarrow{a} = \frac{d\overrightarrow{v}}{dt} $ Since velocity flips sign ($\overrightarrow{v} \to -\overrightarrow{v}$), acceleration also flips sign: $ \overrightarrow{a'} = \frac{d(-\overrightarrow{v})}{dt} = -\frac{d\overrightarrow{v}}{dt} = -\overrightarrow{a} $ Acceleration flips sign.
Linear Momentum ($\overrightarrow{p}$)
Linear momentum is the product of mass (\(m\)) and velocity: $ \overrightarrow{p} = m\overrightarrow{v} $ Assuming mass \(m\) remains constant, if velocity flips sign, linear momentum also flips sign: $ \overrightarrow{p'} = m\overrightarrow{v'} = m(-\overrightarrow{v}) = -\overrightarrow{p} $ Linear momentum flips sign.
Angular Momentum ($\overrightarrow{L}$)
Angular momentum is defined as the cross product of the position vector and linear momentum: $ \overrightarrow{L} = \overrightarrow{r} \times \overrightarrow{p} $ Under the sign change transformation ($\overrightarrow{r} \to -\overrightarrow{r}$ and $\overrightarrow{p} \to -\overrightarrow{p}$): $ \overrightarrow{L'} = (-\overrightarrow{r}) \times (-\overrightarrow{p}) $ Using the properties of the cross product, $(-a\vec{x}) \times (-b\vec{y}) = ab(\vec{x} \times \vec{y})$: $ \overrightarrow{L'} = (-1)(-1) (\overrightarrow{r} \times \overrightarrow{p}) = \overrightarrow{L} $ Angular momentum does not flip sign.
Conclusion
The analysis demonstrates that linear momentum, velocity, and acceleration change sign when the position vector $\overrightarrow{r}$ changes sign. Angular momentum does not change sign.
The question asks which vector will not flip sign. Based on the provided correct answer, Velocity is the selected option.
Final Answer: The final answer is $\boxed{\text{Velocity}}$
Similar Questions
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Important Questions from General Physics
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A man in a car at location Q on a straight highway is moving with speed v. He decides to reach a point P in qa field at a distance d from the highway (point m) as shown in the figure. Speed of the car in the field is half to that on the highway. What should be the distance RM, so that the time taken to reach P is minimum ?
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