3.4 Motion in space  (Page 2/17)

To gain a better understanding of the velocity and acceleration vectors, imagine you are driving along a curvy road. If you do not turn the steering wheel, you would continue in a straight line and run off the road. The speed at which you are traveling when you run off the road, coupled with the direction, gives a vector representing your velocity, as illustrated in the following figure.

However, the fact that you must turn the steering wheel to stay on the road indicates that your velocity is always changing (even if your speed is not) because your direction is constantly changing to keep you on the road. As you turn to the right, your acceleration vector also points to the right. As you turn to the left, your acceleration vector points to the left. This indicates that your velocity and acceleration vectors are constantly changing, regardless of whether your actual speed varies ( [link] ).

Components of the acceleration vector

We can combine some of the concepts discussed in Arc Length and Curvature with the acceleration vector to gain a deeper understanding of how this vector relates to motion in the plane and in space. Recall that the unit tangent vector T and the unit normal vector N form an osculating plane at any point P on the curve defined by a vector-valued function $\text{r}\left(t\right).$ The following theorem shows that the acceleration vector $\text{a}\left(t\right)$ lies in the osculating plane and can be written as a linear combination of the unit tangent and the unit normal vectors.

Proof

Because $\text{v}\left(t\right)=r^{\prime }\left(t\right)$ and $\text{T}\left(t\right)=\frac{r^{\prime }\left(t\right)}{\Vert r^{\prime }\left(t\right)\Vert },$ we have $\text{v}\left(t\right)=\Vert r^{\prime }\left(t\right)\Vert \text{T}\left(t\right)=v\left(t\right)\text{T}\left(t\right).$ Now we differentiate this equation:

Since $\text{N}\left(t\right)=\frac{T^{\prime }\left(t\right)}{\Vert T^{\prime }\left(t\right)\Vert },$ we know $T^{\prime }\left(t\right)=\Vert T^{\prime }\left(t\right)\Vert \text{N}\left(t\right),$ so

A formula for curvature is $\kappa =\frac{\Vert T^{\prime }\left(t\right)\Vert }{\Vert r^{\prime }\left(t\right)\Vert },$ so $\Vert T^{\prime }\left(t\right)\Vert =\kappa \Vert r^{\prime }\left(t\right)\Vert =\kappa v\left(t\right).$ This gives $\text{a}\left(t\right)=v^{\prime }\left(t\right)\text{T}\left(t\right)+\kappa {\left(v\left(t\right)\right)}^2\text{N}\left(t\right).$

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The coefficients of $\text{T}\left(t\right)$ and $\text{N}\left(t\right)$ are referred to as the tangential component of acceleration    and the normal component of acceleration    , respectively. We write $a_{\text{T}}$ to denote the tangential component and $a_{\text{N}}$ to denote the normal component.