Problem 46 - Thomas Calculus in SI Units

Question

Give the acceleration \(a=d^{2} s / d t^{2},\) initial velocity, and initial position of an object moving on a coordinate line. Find the object's position at time \(t\). $$a=9.8, \quad v(0)=-3, \quad s(0)=0$$

Step-by-Step Solution

Answer

The object's position at time \(t\) is \(s(t) = 4.9t^2 - 3t\).

Step 1: Integrate acceleration to get velocity

Given the acceleration \(a = 9.8\). To find the velocity function \(v(t)\), we need to integrate the acceleration with respect to time \(t\). Thus, \(v(t) = \int 9.8 \, dt = 9.8t + C_1\).

Step 2: Use initial velocity to find constant

We know that \(v(0) = -3\). Substitute \(t = 0\) into the velocity equation and solve for \(C_1\): \[-3 = 9.8(0) + C_1 \Rightarrow C_1 = -3\].Therefore, the velocity function is \(v(t) = 9.8t - 3\).

Step 3: Integrate velocity to get position

Now integrate \(v(t) = 9.8t - 3\) to find the position function \(s(t)\). So, \[s(t) = \int (9.8t - 3) \, dt = 4.9t^2 - 3t + C_2\].

Step 4: Use initial position to find constant

Given \(s(0) = 0\). Substitute \(t = 0\) into the position equation and solve for \(C_2\): \[0 = 4.9(0)^2 - 3(0) + C_2 \Rightarrow C_2 = 0\].So, the position function is \(s(t) = 4.9t^2 - 3t\).

Key Concepts

Understanding Acceleration

Acceleration is a measure of how quickly an object changes its velocity. In simple terms, it's the rate at which velocity increases or decreases over time. If you've ever felt a car speed up, you've experienced acceleration firsthand.
\[a(t) = \frac{d^2s}{dt^2}\]
Here, \(a(t)\) represents the acceleration, and it's the second derivative of the position function \(s(t)\) with respect to time \(t\). The higher the acceleration, the faster the velocity changes.

• A positive acceleration means the object is speeding up.
• A negative acceleration, often called deceleration, means the object is slowing down.

In our problem, the acceleration is constant at 9.8 m/s², similar to the gravitational pull on Earth. Understanding this helps predict how an object's speed will change over a given period.

The Role of Velocity

Velocity describes how fast an object is moving in a particular direction. It's a vector quantity, which means it has both magnitude (speed) and direction.
\[v(t) = \int a(t) \, dt = 9.8t + C_1\]
When dealing with calculus, velocity is the first derivative of the position function. In simpler terms, it tells us how the position changes over time.

• Initial velocity is the starting speed of the object at time \(t = 0\).
• Negative velocity indicates movement in the opposite direction to the positive coordinate direction.

In the given exercise, the initial velocity is -3 m/s. This indicates the object starts moving in the opposite direction of the positive axis line.

Grasping the Position Function

The position function, typically denoted as \(s(t)\), provides the location of an object at any given time. It's an essential tool to predict where an object will be on a line or in space. For instance, in our exercise:
\[s(t) = \int v(t) \, dt = 4.9t^2 - 3t + C_2\]
This function results from integrating the velocity function. Its practical use is in scenarios like computing where a car will be after a certain drive time given its current speed and acceleration.

• The initial position, given by \(s(0)\), helps determine the constant of integration (\(C_2\)).
• This constant reflects any starting point offset in the motion scenario.

For the problem here, the initial position \(s(0) = 0\) simplifies our position equation by removing \(C_2\) as it equals zero.

Power of Integration in Motion Analysis

Integration plays a vital role in determining an object's motion characteristics when acceleration is involved. It's the mathematical process of finding the function whose derivative is the given function, which is crucial in retrieving velocity and position from acceleration data.
In calculus:

• The integral of an acceleration function gives the velocity function, providing insights into how speed changes over time.
• Further integrating the velocity function yields the position function, offering a tool to predict the specific location of an object in its path.

By integrating, we connect these changes across time, forming a comprehensive picture of motion. This becomes especially powerful when acceleration varies, making it indispensable in areas like physics and engineering to make accurate predictions of future positions.