Question

Find \(v(t), a(t), \mathbf{T}(t),\) and \(N(t)\) (if it exists) for an object moving along the path given by the vector-valued function \(\mathrm{r}(t) .\) Use the results to determine the form of the path. Is the speed of the object constant or changing? $$ \mathbf{r}(t)=4 t^{2} \mathbf{i} $$

Short answer

Objects' velocity and acceleration are \(8t\mathbf{i}\) and \(8\mathbf{i}\) respectively. The unit tangent vector is \(\mathrm{sgn}(t)\mathbf{i}\) and the normal vector doesn't exist due to the straight-line motion. The object follows a straight-line path along the x-axis with a non-constant speed of \(8|t|\).

Step-by-step solution

Find the Velocity

The velocity \(\mathbf{v}(t)\) is the first derivative of the position vector \(\mathbf{r}(t)\). This means, \(\mathbf{v}(t)=\mathbf{r}^{\prime}(t) = \frac{d}{dt} (4t^{2}\mathbf{i}) = 8t\mathbf{i}\).

Find the Acceleration

The acceleration \(\mathbf{a}(t)\) is the derivative of the velocity vector \(\mathbf{v}(t)\). Therefore \(\mathbf{a}(t)=\mathbf{v}^{\prime}(t) = \frac{d}{dt} (8t\mathbf{i}) = 8\mathbf{i}\).

Calculate the Unit Tangent Vector

The unit tangent vector \(\mathbf{T}(t)\) is the velocity vector divided by its magnitude. The magnitude of the velocity vector, \(|\mathbf{v}(t)|\) is calculated as \( \sqrt{(8t)^2}=8|t|\). Therefore, \(\mathbf{T}(t)= \frac{\mathbf{v}(t)}{|\mathbf{v}(t)|} = \frac{8t\,\mathbf{i}}{8|t|}=\mathrm{sgn}(t)\mathbf{i}\), where sgn(t) is the signum function.

Find the Unit Normal Vector

For a straight-line motion in one direction, the unit normal vector doesn't exist because there is no change in direction.

Determine the Form of the Path

According to the provided vector-valued function \(\mathbf{r}(t)=4t^{2}\mathbf{i}\), we can infer that the object's motion is a straight line in the direction of the \(\mathbf{i}\) vector which is along x-axis.

Check if the Speed is Constant or Changing

The speed at any time \(t\) is the magnitude of the velocity vector \(\mathbf{v}(t)\), i.e. \(|\mathbf{v}(t)|= 8|t|\), which is a function of \(t\). This indicates that the speed of the object is changing over time, it is not constant.