Question

The function \(s(t)\) represents the position of an object at time \(t\) moving along a line. Suppose \(s(1)=84\) and \(s(4)=144 .\) Find the average velocity of the object over the interval of time [1,4]

Short answer

Answer: The average velocity of the object over the interval of time [1, 4] is 20.

Step-by-step solution

Find the difference in position

To find the difference in position, subtract the initial position from the final position: \(s(4) - s(1) = 144 - 84 = 60\).

Find the difference in time

To find the difference in time, subtract the initial time from the final time: \(4 - 1 = 3\).

Calculate the average velocity

The average velocity is the difference in position divided by the difference in time: \(\frac{s(4) - s(1)}{4 - 1} = \frac{60}{3} = 20\). The average velocity of the object over the interval of time [1,4] is 20.

Key concepts

Position-Time Relationship

When studying physics and calculus, one fundamental concept to understand is the 'position-time relationship' in motion problems. This relationship tells us how the position of an object changes with time.

Consider the provided exercise, we have a function, denoted as \( s(t) \), which tells us the position of an object at any given time \( t \). To visualize this relationship, imagine plotting \( s(t) \) on a graph where the x-axis represents time and the y-axis represents position. The shape of the curve will show us how the object is moving - whether it's moving forward or backward, accelerating or decelerating.

Understanding the graph's slope at any point gives us the instantaneous velocity at that time. Moreover, when we're interested in the average velocity over a period, as in the textbook exercise, we look at the overall change in position over the total time elapsed. This relationship is key to solving many kinds of motion problems using calculus.

Average Velocity Calculation

Calculating average velocity is a crucial skill in physics and calculus. It's defined as the total displacement divided by the total time taken. Here's a straightforward formula for finding average velocity, \( v_{avg} \):

\[ v_{avg} = \frac{\Delta s}{\Delta t} \]

In this formula, \( \Delta s \) represents the change in position, while \( \Delta t \) is the change in time. Remember, displacement can be positive or negative, which will affect the sign of the average velocity, indicating direction. From our exercise, we take the positions at two different times, and by substituting these values into our formula, like so:

\[ v_{avg} = \frac{s(4) - s(1)}{4 - 1} = \frac{144 - 84}{3} = 20 \]

We find that our object has an average velocity of 20 units per time interval. This calculation is fundamental in analyzing the movement of objects and is widely used in various physics problems.

Calculus Motion Problems

Motion problems in calculus often involve variables such as position, velocity, and acceleration, each related to time. To tackle these problems, it's crucial to understand derivatives and integrals as they relate to motion.

The position function, such as \( s(t) \) in our exercise, can be differentiated to find the velocity function \( v(t) \), and then differentiated again to find the acceleration function \( a(t) \). Conversely, if you have the acceleration, you can integrate to find velocity, and then integrate again to find position, possibly with the aid of initial conditions.

In our exercise, we're working with a snapshot of the position function's behavior over a certain interval. But in more complex problems, you may be asked to find instantaneous velocity (the derivative at a certain point) or the object's position at a different time (using the integral of the velocity). These types of problems are essential for students to master as they provide a foundation for understanding real-world physics scenarios and other advanced topics in calculus.