Question

# Explain how a particle can be accelerating even though its speed is constant.

Solution

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$\hspace*{5mm}$Position vector is

$\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}$

$\hspace*{5mm}$The velocity vector is

\begin{align*} \mathbf{v}(t) &=\mathbf{r}^{\prime}(t) \\ &=-\sin t \mathbf{i}+\cos t \mathbf{j} \end{align*}

$\hspace*{5mm}$Speed is

\begin{align*} \boldsymbol{v} &=\|\mathbf{v}(t)\| \\ &=\sqrt{(-\sin t)^{2}+(\cos t)^{2}} \\ &=1 \end{align*}

$\hspace*{5mm}$From this we can see that speed is constant, and acceleration vector is $\mathbf{a}(t)=-\cos t \mathbf{i}-\sin t \mathbf{j}$ and this change all time.

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