Question

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

Solution

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\hspace*{5mm}Position vector is

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

\hspace*{5mm}The velocity vector is

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

\hspace*{5mm}Speed is

v=v(t)=(sint)2+(cost)2=1\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 a(t)=costisintj\mathbf{a}(t)=-\cos t \mathbf{i}-\sin t \mathbf{j} and this change all time.

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