Question

Given the position function \(\mathbf{r}\) of a moving object, explain how to find the velocity, speed, and acceleration of the object.

Short answer

Answer: To find the velocity, speed, and acceleration of a moving object with position function \(\mathbf{r}(t)\), follow these steps: 1. Find the velocity \(\mathbf{v}(t)\) by taking the first derivative of the position function with respect to time. 2. Find the speed \(v(t)\) by calculating the magnitude of the velocity vector. 3. Find the acceleration \(\mathbf{a}(t)\) by taking the first derivative of the velocity vector (or second derivative of the position function) with respect to time.

Step-by-step solution

Find the velocity

To find the velocity of the object, we need to find the first derivative of the position function \(\mathbf{r}(t)\). The first derivative \(\mathbf{v}(t)\) represents the rate of change of the position function, which is the velocity. To calculate the velocity, take the derivative of each component in the position function with respect to time, \(t\): $$\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt}$$

Find the speed

Speed is the magnitude of the velocity vector. To find the speed, we calculate the magnitude of the velocity vector \(\mathbf{v}(t)\), usually denoted as \(|\mathbf{v}(t)|\) or \(v(t)\). Let's assume that the velocity vector has components \(\mathbf{v}(t) = (v_x(t), v_y(t), v_z(t))\). Then the speed \(v(t)\) can be calculated using the Pythagorean theorem for 3 dimensions: $$v(t) = |\mathbf{v}(t)| = \sqrt{v_x(t)^2 + v_y(t)^2 + v_z(t)^2}$$

Find the acceleration

The acceleration of the object is the rate of change of the velocity vector \(\mathbf{v}(t)\). Therefore, we need to find the first derivative of the velocity vector, which is the second derivative of the position function \(\mathbf{r}(t)\). The resulting vector \(\mathbf{a}(t)\) represents the acceleration of the object: $$\mathbf{a}(t) = \frac{d^2\mathbf{r}(t)}{dt^2} = \frac{d\mathbf{v}(t)}{dt}$$ To summarize, given the position function \(\mathbf{r}(t)\) of a moving object: 1. Find the velocity \(\mathbf{v}(t)\) by taking the first derivative of the position function with respect to time. 2. Find the speed \(v(t)\) by calculating the magnitude of the velocity vector. 3. Find the acceleration \(\mathbf{a}(t)\) by taking the first derivative of the velocity vector (or second derivative of the position function) with respect to time.