Question
What is the velocity of a particle given its vector position expression?
Answers
Answered by
GPT-4o mini
To find the velocity of a particle given its vector position expression, you need to differentiate the position vector with respect to time.
If the position vector \(\mathbf{r}(t)\) is given in the form:
\[
\mathbf{r}(t) = x(t) \hat{i} + y(t) \hat{j} + z(t) \hat{k}
\]
where \(x(t)\), \(y(t)\), and \(z(t)\) are functions of time \(t\), you can find the velocity \(\mathbf{v}(t)\) by taking the derivative:
\[
\mathbf{v}(t) = \frac{d\mathbf{r}}{dt} = \frac{dx}{dt} \hat{i} + \frac{dy}{dt} \hat{j} + \frac{dz}{dt} \hat{k}
\]
This gives you the velocity vector in terms of its components, which are the derivatives of the position functions with respect to time.
In summary, the velocity \(\mathbf{v}(t)\) of the particle is given by:
\[
\mathbf{v}(t) = \dot{x}(t) \hat{i} + \dot{y}(t) \hat{j} + \dot{z}(t) \hat{k}
\]
where \(\dot{x}(t) = \frac{dx}{dt}\), \(\dot{y}(t) = \frac{dy}{dt}\), and \(\dot{z}(t) = \frac{dz}{dt}\).
If the position vector \(\mathbf{r}(t)\) is given in the form:
\[
\mathbf{r}(t) = x(t) \hat{i} + y(t) \hat{j} + z(t) \hat{k}
\]
where \(x(t)\), \(y(t)\), and \(z(t)\) are functions of time \(t\), you can find the velocity \(\mathbf{v}(t)\) by taking the derivative:
\[
\mathbf{v}(t) = \frac{d\mathbf{r}}{dt} = \frac{dx}{dt} \hat{i} + \frac{dy}{dt} \hat{j} + \frac{dz}{dt} \hat{k}
\]
This gives you the velocity vector in terms of its components, which are the derivatives of the position functions with respect to time.
In summary, the velocity \(\mathbf{v}(t)\) of the particle is given by:
\[
\mathbf{v}(t) = \dot{x}(t) \hat{i} + \dot{y}(t) \hat{j} + \dot{z}(t) \hat{k}
\]
where \(\dot{x}(t) = \frac{dx}{dt}\), \(\dot{y}(t) = \frac{dy}{dt}\), and \(\dot{z}(t) = \frac{dz}{dt}\).