Tangential Component of Acceleration Calculator

Find the tangential component of acceleration for $$$\mathbf{\vec{r}\left(t\right)} = \left\langle t, t^{2}, t^{3}\right\rangle$$$.

Find the derivative of $$$\mathbf{\vec{r}\left(t\right)}$$$: $$$\mathbf{\vec{r}^{\prime}\left(t\right)} = \left\langle 1, 2 t, 3 t^{2}\right\rangle$$$ (for steps, see derivative calculator).

Find the magnitude of $$$\mathbf{\vec{r}^{\prime}\left(t\right)}$$$: $$$\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\right\rvert} = \sqrt{9 t^{4} + 4 t^{2} + 1}$$$ (for steps, see magnitude calculator).

Find the derivative of $$$\mathbf{\vec{r}^{\prime}\left(t\right)}$$$: $$$\mathbf{\vec{r}^{\prime\prime}\left(t\right)} = \left\langle 0, 2, 6 t\right\rangle$$$ (for steps, see derivative calculator).

Find the dot product: $$$\mathbf{\vec{r}^{\prime}\left(t\right)}\cdot \mathbf{\vec{r}^{\prime\prime}\left(t\right)} = 18 t^{3} + 4 t$$$ (for steps, see dot product calculator).

Finally, the tangential component of acceleration is $$$a_T\left(t\right) = \frac{\mathbf{\vec{r}^{\prime}\left(t\right)}\cdot \mathbf{\vec{r}^{\prime\prime}\left(t\right)}}{\mathbf{\left\lvert \mathbf{\vec{r}^{\prime}\left(t\right)}\right\rvert}} = \frac{18 t^{3} + 4 t}{\sqrt{9 t^{4} + 4 t^{2} + 1}}.$$$

The tangential component of acceleration is $$$a_T\left(t\right) = \frac{18 t^{3} + 4 t}{\sqrt{9 t^{4} + 4 t^{2} + 1}}$$$A.