Question

The position vector \(r\) describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. $$ \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\frac{t^{2}}{2} \mathbf{k} $$

Short answer

The velocity of the object is \(v(t) = \mathbf{i} + 2t \mathbf{j} + t \mathbf{k}\), the speed at time \(t\) is \(\sqrt{1 + 4t^2 + t^2}\), and acceleration is \(a(t) = 2\mathbf{j} + \mathbf{k}\)

Step-by-step solution

Calculate the Velocity

Velocity vector is the first derivative of the position vector with respect to time \(t\). This can be found by differentiating each component of the position vector \(r\) with respect to \(t\), i.e., \(v(t) = dr/dt\). Differentiating each component we get: \(v(t) = \mathbf{i} + 2t \mathbf{j} + t \mathbf{k}\)

Calculate the Speed

Speed is the magnitude of the velocity vector. The magnitude of a vector \(A = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}\) is given as \(\sqrt{{a^2 + b^2 + c^2}}\). Therefore, the speed at time \(t\) is \(\sqrt{1 + 4t^2 + t^2}\)

Calculate the Acceleration

Acceleration vector is the first derivative of the velocity vector with respect to time \(t\). This can be found by differentiating each component of the velocity vector \(v\) with respect to \(t\), i.e., \(a(t) = dv/dt\). Differentiating each component we get: \(a(t) = 2\mathbf{j} + \mathbf{k}\)