Question

Displacement from velocity The following functions describe the velocity of a car (in \(\mathrm{mi} / \mathrm{hr}\) ) moving along a straight highway for a 3-hr interval. In each case, find the function that gives the displacement of the car over the interval \([0, t]\), where \(0 \leq t \leq 3\) (Check your book to see figure) $$v(t)=\left\\{\begin{array}{ll}40 & \text { if } 0 \leq t \leq 1.5 \\\50 & \text { if } 1.5

Short answer

Answer: The displacement function of the car over the interval [0, t] is given by: $$ x(t) = \left\\{\begin{array}{ll}40t & \text { if } 0 \leq t \leq 1.5 \\\50t -15 & \text { if } 1.5<t \leq 3\end{array}\right. $$

Step-by-step solution

Integrate v(t) for the first case (0 ≤ t ≤ 1.5)

Our first case is \(v(t) = 40\) when \(0 \leq t \leq 1.5\). For this case, we integrate the function \(v(t) = 40\) with respect to \(t\) and add a constant \(C_1\) as follows: $$ \int v(t) \, dt = \int 40 \, dt = 40t + C_1 $$

Integrate v(t) for the second case (1.5 < t ≤ 3)

Our second case is \(v(t) = 50\) when \(1.5 < t \leq 3\). For this case, we integrate the function \(v(t) = 50\) with respect to \(t\) and add a constant \(C_2\) as follows: $$ \int v(t) \, dt = \int 50 \, dt = 50t + C_2 $$

Find C1 (apply continuity condition)

To find \(C_1\), we'll apply the continuity condition of the displacement function at the point where the two cases of the velocity function change, i.e., when \(t = 1.5\). We'll set both the displacement functions from Steps 1 and 2 equal to each other at \(t = 1.5\): $$ 40(1.5) + C_1 = 50(1.5) + C_2 $$

Solve for C1 and C2

We want to find the values for \(C_1\) and \(C_2\). In Step 3, we got the equation: $$ 40(1.5) + C_1 = 50(1.5) + C_2 $$ This simplifies to: $$ 60 + C_1 = 75 + C_2 $$ We can see that \(C_2 = C_1 - 15\). Since \(x = 0\) is the starting point of the motion, it makes sense to set the displacement \(x(0) = 0\) too. Let's now substitute \(t = 0\) in the equation for the first displacement function (from Step 1): $$ x(0) = 40(0) + C_1 = 0 $$ We find that \(C_1 = 0\). Therefore, \(C_2 = 0 - 15 = -15\).

Write the displacement function

Considering the values of \(C_1\) and \(C_2\) found in Step 4, we can now write the piecewise displacement function as follows: $$ x(t) = \left\\{\begin{array}{ll}40t & \text { if } 0 \leq t \leq 1.5 \\\50t -15 & \text { if } 1.5<t \leq 3\end{array}\right. $$ This is the displacement function of the car over the interval \([0, t]\), where \(0 \leq t \leq 3\).

Key concepts

Integration of Velocity

The concept of integration of velocity is fundamental when studying motion. Integrating the velocity function gives us the displacement, the total distance covered by an object in a certain direction. In a practical sense, when we integrate a velocity function v(t) over time t, we are essentially summing up tiny segments of distance over a period, which provides us with the overall change in position.

For a constant velocity, the integration process is straightforward. For instance, if a car moves at a constant velocity of 40 mph, the displacement after t hours is simply 40 times t, plus any initial displacement. However, if the velocity changes over time, we may need to perform separate integrations over different intervals, and this is where the concept of a piecewise function becomes important.

Piecewise Function

A piecewise function is a tool that allows us to describe a function that behaves differently over various intervals within its domain. In our context, the velocity of the car is given by a piecewise function with different values for different time intervals.

These types of functions are useful in practical situations where a sudden change in the behavior of a function occurs, such as a car accelerating or decelerating. The velocity function provided in the exercise is a classic example of a piecewise function with two separate constant velocities over two distinct time intervals. To determine the displacement function, we need to integrate the velocity function over each interval and then use the initial conditions to ensure that the entire function is continuous.

Initial Conditions

In the context of motion, initial conditions tell us about the starting point of the object. They are essential for solving problems involving differentiation or integration because they allow us to determine any constants of integration that arise. The displacement function generally has the form x(t) = vt + C, where C is a constant determined by the initial condition.

When an object begins its journey from the origin or a known position, this information can be used to find out what the constant C is when t = 0. In our problem, we apply the condition that at t = 0, the displacement is 0, which led to the solution of the constants for both sections of the piecewise function and ensured that the overall displacement function is continuous and accurate.

Velocity Function

The velocity function, often denoted as v(t), represents the rate of change of the position of an object with respect to time. It is a pivotal concept in the study of kinematics, the branch of physics that deals with motion. In our exercise, the velocity function is provided as a simple piecewise function with two constant values over different time intervals.

Understanding the velocity function is crucial because it leads to the calculation of displacement through integration. Also, the shape and behavior of the velocity function can provide insights into the dynamics of the object's motion, telling us if it's moving at a constant speed, accelerating, or decelerating over time. The shape of the velocity function might be more complex in other contexts, requiring more in-depth integration techniques, reflecting real-world scenarios where the speed of an object changes in a more complicated manner.