Problem 10 - Calculus Early Transcendentals

Question

Average velocity The position of an object moving vertically along a line is given by the function \(s(t)=-4.9 t^{2}+30 t+20\) Find the average velocity of the object over the following intervals. a. [0,3] b. [0,2] c. [0,1] d. \([0, h],\) where \(h>0\) is a real number.

Step-by-Step Solution

Answer

Answer: The average velocities for the given time intervals are: a. For the interval [0, 3], the average velocity is 15.6 m/s. b. For the interval [0, 2], the average velocity is 12.8 m/s. c. For the interval [0, 1], the average velocity is 25.1 m/s. d. For any interval [0, h] with \(h > 0\), the average velocity is \(30-9.8h \ \frac{m}{s}\).

Step 1: Calculate s(t) for each interval

Calculating final and initial positions for each interval. a. [0,3]: \(s(0)=-4.9(0)^{2}+30(0)+20=20\) and \(s(3)=-4.9(3)^{2}+30(3)+20 = -4.9(9)+90+20=66.9\) b. [0,2]: \(s(2)=-4.9(2)^{2}+30(2)+20 = -4.9(4)+60+20=45.6\) c. [0,1]: \(s(1)=-4.9(1)^{2}+30(1)+20 = -4.9+30+20=45.1\) d. \([0, h]\): \(s(h)=-4.9h^{2}+30h+20\)

Step 2: Calculate average velocities for each interval

Calculating average velocities using the formula \(v_{avg}=\frac{s(t_2)-s(t_1)}{t_2-t_1}\). a. \(v_{avg}=\frac{s(3)-s(0)}{3-0}=\frac{66.9-20}{3}=15.6\frac{m}{s}\) b. \(v_{avg}=\frac{s(2)-s(0)}{2-0}=\frac{45.6-20}{2}=12.8\frac{m}{s}\) c. \(v_{avg}=\frac{s(1)-s(0)}{1-0}=\frac{45.1-20}{1}=25.1\frac{m}{s}\) d. \(v_{avg}=\frac{s(h)-s(0)}{h-0}=\frac{-4.9h^{2}+30h+20-20}{h}=30-9.8h \frac{m}{s}\) The average velocities are: a. 15.6 m/s b. 12.8 m/s c. 25.1 m/s d. \(30-9.8h \ \frac{m}{s}\) for \(h > 0\)

Key Concepts

Kinematics in Calculus

Kinematics is the study of motion, describing the trajectory of an object as it moves through space. Calculus, especially differential calculus, plays a crucial role in understanding and analyzing this motion mathematically. It deals with variables that continuously change, such as the position of an object over time. In kinematics, we can use calculus to find velocity, acceleration, and other aspects of an object's motion by deriving the position-time function or integrating the acceleration-time function. These calculations are indispensable in physics, engineering, and any field that deals with moving systems.

Average Velocity Formula

The average velocity of an object is defined as the total displacement divided by the total time taken for that displacement. Mathematically, we use the formula \( v_{avg} = \frac{\Delta s}{\Delta t} = \frac{s(t_2)-s(t_1)}{t_2-t_1} \), where \( s(t) \) is the position at time \( t \), and \( t_1 \) and \( t_2 \) represent the initial and final times, respectively. This formula provides us with a scalar quantity that gives the overall rate at which position changes. Average velocity can be a starting point for understanding more intricate motion patterns, and it's a fundamental concept both in basic and advanced kinematics.

Position-Time Function

The position-time function in kinematics, represented as \( s(t) \), denotes the position of an object at any given time \( t \). It's a mathematical description of an object's motion, usually derived from the principles of physics and initial conditions. Through differentiation and integration, these functions allow us to find velocity, acceleration, and other characteristics of motion at specific moments. These applications are not only theoretical but also serve practical purposes such as predicting the position of satellites, planning the trajectory of vehicles, and much more.

Role in Average Velocity

When dealing with average velocity, the position-time function is essential, as it is from this function that we extract the initial and final positions required to calculate the average velocity over a given time interval.

Calculus Application in Motion

Calculus is integral to studying motion because it allows us to deal with changes that occur over continuous intervals. When an object's position is described by a function of time, differentiation gives us the rate of position change, which is velocity. When we take the derivative of the velocity function, we get acceleration, which tells us how the speed of an object is changing. Integration, on the other hand, takes us from acceleration back to velocity, and from velocity back to position. These operations are essential for analyzing real-world motion scenarios and for making predictions about an object's future state.

Using Calculus to Improve Understanding

By applying calculus to the position-time function, students can gain a deeper understanding of motion, connecting physical concepts with mathematical techniques. For example, differentiating the position-time function provides instantaneous velocity, while integrating the acceleration function yields velocity over time. This application not only enhances comprehension but also equips students with tools to solve complex problems in kinematics and beyond.