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Finding the Velocity, Acceleration, and Speed of a Vector Particle in 3D (Example 1)
You may be asked to find velocity, acceleration, or speed of a moving particle, where the path of a particle is represented by a curvature equation r(t).
The key to this is just finding derivatives or integrating.
Sample question: Find the velocity, acceleration, and speed of a particle.
The particle’s position function is given by the vector function \(r(t) = \left\langle { – \frac{1}{2}{t^2},t} \right\rangle \)
The velocity is just the differentiation of the position function, and the acceleration is the second derivative of the position function.
The speed is just the velocity function’s norm (you remove the directional components from the velocity equation by finding the magnitude/norm).
The norm is just the square of each component, summed together, then you take the square root of that.
First we can find the velocity function v(t). To differentiate a vector function just differentiate each component.
\[r(t) = \left\langle { – \frac{1}{2}{t^2},t} \right\rangle \]
\[r'(t) = v(t) = \left\langle { – t,1} \right\rangle \]
And the speed function is just the norm of this.
\[\left| {r'(t)} \right| = \left| {v(t)} \right| = \sqrt {{t^2} + 1} \]
And acceleration is just the differentiation of the velocity function (second derivative of the position function):
\[r”(t) = a(t) = \left\langle { – 1,0} \right\rangle \]
Done!
Another variation of this question is to give the acceleration function and ask you to find the position function, so in that case you’d just work backwards by integrating and using given initial conditions.
Click here to see another harder example with a moving particle equation!