Do the dot product of the velocity and acceleration above

April 13, 2026Physics

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Answer

\vec{v}(t) \cdot \vec{a}(t) = 0

3 steps

To find the dot product of the velocity and acceleration vectors, we multiply their corresponding components and sum the results.

From the previous steps:
The velocity vector is:

\vec{v}(t) = \frac{1}{2}\left(-\sin\left(\frac{t}{2}\right)\hat{i} + \cos\left(\frac{t}{2}\right)\hat{j}\right)

The acceleration vector is:

\vec{a}(t) = -\frac{1}{4}\left(\cos\left(\frac{t}{2}\right)\hat{i} + \sin\left(\frac{t}{2}\right)\hat{j}\right)

Identify the components of each vector.
For $\vec{v}(t)$ :
$v_x = -\frac{1}{2}\sin\left(\frac{t}{2}\right)$
$v_y = \frac{1}{2}\cos\left(\frac{t}{2}\right)$

For $\vec{a}(t)$ :
$a_x = -\frac{1}{4}\cos\left(\frac{t}{2}\right)$
$a_y = -\frac{1}{4}\sin\left(\frac{t}{2}\right)$
Calculate the dot product $\vec{v}(t) \cdot \vec{a}(t) = v_x a_x + v_y a_y$ .
$\vec{v}(t) \cdot \vec{a}(t) = \left(-\frac{1}{2}\sin\left(\frac{t}{2}\right)\right)\left(-\frac{1}{4}\cos\left(\frac{t}{2}\right)\right) + \left(\frac{1}{2}\cos\left(\frac{t}{2}\right)\right)\left(-\frac{1}{4}\sin\left(\frac{t}{2}\right)\right)$
Simplify the expression.
$\vec{v}(t) \cdot \vec{a}(t) = \frac{1}{8}\sin\left(\frac{t}{2}\right)\cos\left(\frac{t}{2}\right) - \frac{1}{8}\cos\left(\frac{t}{2}\right)\sin\left(\frac{t}{2}\right)$ $\vec{v}(t) \cdot \vec{a}(t) = 0$

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