Problem 17 - Essential Calculus Early Transcendentals

Question

A force with magnitude 20 \(\mathrm{N}\) acts directly upward from the \(x y\) -plane on an object with mass 4 \(\mathrm{kg}\) . The object starts at the origin with initial velocity \(\mathbf{v}(0)=\mathbf{i}-\mathbf{j} .\) Find its position function and its speed at time \(t\)

Step-by-Step Solution

Answer

Position function: \(\mathbf{r}(t) = t\mathbf{i} - t\mathbf{j} + \frac{5t^2}{2} \mathbf{k}\); Speed: \(\sqrt{2 + 25t^2}\).

Step 1: Define the Problem

The object experiences a force of 20 N directly upward (along the z-axis) and has an initial velocity vector \(\mathbf{v}(0) = \mathbf{i} - \mathbf{j}\). We need to find its position function and speed at time \(t\).

Step 2: Determine Acceleration

Using Newton's second law \(\mathbf{F} = m\mathbf{a}\), we calculate the acceleration. The force is 20 N upward on a 4 kg object, so the acceleration is \( \mathbf{a} = \frac{20}{4} \mathbf{k} = 5 \mathbf{k} \text{ m/s}^2\).

Step 3: Find Velocity Function

The velocity function is found by integrating the acceleration. Since \(\mathbf{a} = 5 \mathbf{k}\), the velocity is \(\mathbf{v}(t) = \int 5 \mathbf{k} \, dt = 5t \mathbf{k} + \mathbf{C}\), where \(\mathbf{C}\) is the constant vector of integration. Given \(\mathbf{v}(0) = \mathbf{i} - \mathbf{j}\), \(\mathbf{C}\) is \(\mathbf{i} - \mathbf{j}\). Thus, \(\mathbf{v}(t) = \mathbf{i} - \mathbf{j} + 5t \mathbf{k}\).

Step 4: Find Position Function

Integrate the velocity function to find the position. \(\mathbf{r}(t) = \int (\mathbf{i} - \mathbf{j} + 5t \mathbf{k}) \, dt = t\mathbf{i} - t\mathbf{j} + \frac{5t^2}{2} \mathbf{k} + \mathbf{D}\). Since the initial position is \(\mathbf{r}(0) = \mathbf{0}\), \(\mathbf{D} = \mathbf{0}\). Hence, \(\mathbf{r}(t) = t\mathbf{i} - t\mathbf{j} + \frac{5t^2}{2} \mathbf{k}\).

Step 5: Calculate Speed at Time \(t\)

Speed is the magnitude of the velocity vector. So, \(\text{Speed} = ||\mathbf{v}(t)|| = ||\mathbf{i} - \mathbf{j} + 5t \mathbf{k}|| = \sqrt{1^2 + (-1)^2 + (5t)^2} = \sqrt{2 + 25t^2}\).

Key Concepts

Newton's Second Law

Newton's Second Law is a fundamental principle in physics that provides the relationship between the force acting on an object, its mass, and its acceleration. This law is succinctly expressed in the equation \(\mathbf{F} = m\mathbf{a}\), where \(\mathbf{F}\) is the force vector, \(m\) is the mass of the object, and \(\mathbf{a}\) is the acceleration vector.
This principle tells us that the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass. As seen in the problem, a force of 20 N is applied directly upward on a 4 kg object.
According to Newton's Second Law, we can find the acceleration by rearranging the formula: \(\mathbf{a} = \frac{\mathbf{F}}{m}\). Calculating this with our values gives us \(\mathbf{a} = \frac{20}{4} \mathbf{k} = 5 \mathbf{k} \text{ m/s}^2\). This equation helps us understand how the object's motion will change over time.

Velocity Function

The velocity function helps us describe how the object's velocity changes over time. Velocity is essentially a vector quantity that includes both the speed and direction of the object's motion.
Given the constant acceleration found using Newton's Second Law, the velocity function can be determined by integrating the acceleration function.
In our example, the acceleration is \(5 \mathbf{k} \text{ m/s}^2\). By integrating this with respect to time \(t\), we get:

• \(\mathbf{v}(t) = \int 5 \mathbf{k} \, dt = 5t \mathbf{k} + \mathbf{C}\)

Here, \(\mathbf{C}\) is the constant of integration representing the initial velocity of the object. The initial velocity, \(\mathbf{v}(0) = \mathbf{i} - \mathbf{j}\), establishes \(\mathbf{C}\) as \(\mathbf{i} - \mathbf{j}\).
Therefore, the complete velocity function becomes \(\mathbf{v}(t) = \mathbf{i} - \mathbf{j} + 5t \mathbf{k}\). This function shows how the velocity changes as the object moves along different directions over time.

Position Function

The position function describes the object's location in space at any given time \(t\). To find the position function, we integrate the velocity function, as velocity is the rate of change of position.
Our velocity function is \(\mathbf{v}(t) = \mathbf{i} - \mathbf{j} + 5t \mathbf{k}\), and integrating each component independently gives us:

• \(\mathbf{r}(t) = \int (\mathbf{i} - \mathbf{j} + 5t \mathbf{k}) \, dt = t\mathbf{i} - t\mathbf{j} + \frac{5t^2}{2} \mathbf{k} + \mathbf{D}\)

Where \(\mathbf{D}\) is another constant of integration. Given the initial position \(\mathbf{r}(0) = \mathbf{0}\), it follows that \(\mathbf{D} = \mathbf{0}\).
Therefore, the position function is \(\mathbf{r}(t) = t\mathbf{i} - t\mathbf{j} + \frac{5t^2}{2} \mathbf{k}\), which captures the object's trajectory as it moves upwards from the origin.

Acceleration

Acceleration is the rate of change of velocity with respect to time. In our given exercise, the acceleration due to a constant force is represented by the vector \(5 \mathbf{k} \text{ m/s}^2\).
Acceleration is crucial as it tells us how quickly an object's velocity changes, which in turn affects its position over time. This explains the object's motion along the z-axis.
When the acceleration is constant, as in this scenario, the velocity changes uniformly over time. This linear relationship enables simpler calculations of velocity and position by integration.
Constant acceleration situations are common in physics because they allow for straightforward predictions of motion, assuming no other forces act on the object.

Speed Calculation

Speed is a scalar quantity that represents the magnitude of the velocity vector. It is the actual rate at which an object is moving, regardless of direction.
For our object, once we have the velocity function \(\mathbf{v}(t) = \mathbf{i} - \mathbf{j} + 5t \mathbf{k}\), calculating the speed involves finding the magnitude:

• \(\text{Speed} = ||\mathbf{v}(t)|| = \sqrt{1^2 + (-1)^2 + (5t)^2}\)
• \(= \sqrt{2 + 25t^2}\)

This equation gives us a function of \(t\) that tells us how fast the object is moving at any given time.
The speed calculation is essential for understanding the intensity of motion regardless of direction. It allows comparisons of how fast different objects are moving in various directions.