Problem 29 - Calculus Early Transcendentals

Question

Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. $$a(t)=-9.8, v(0)=20, s(0)=0$$

Step-by-Step Solution

Answer

Answer: The position function is \(s(t) = -4.9t^2 + 20t\), and the velocity function is \(v(t) = -9.8t + 20\).

Step 1: Find the velocity function by integrating the acceleration function

The given acceleration function is \(a(t) = -9.8\). Integrate a(t) with respect to t to find the velocity function: $$v(t) = \int a(t) dt = \int -9.8 dt = -9.8t + C.$$ Now, we are given the initial velocity: \(v(0) = 20\). Plug in t=0 into the velocity function to find the constant C: $$20 = -9.8(0) + C \Rightarrow C = 20.$$ So, the velocity function is: $$v(t) = -9.8t + 20.$$

Step 2: Find the position function by integrating the velocity function

Integrate the velocity function \(v(t) = -9.8t + 20\) with respect to t to find the position function: $$s(t) = \int v(t) dt = \int (-9.8t + 20) dt = -4.9t^2 + 20t + D.$$ Now, we are given the initial position: \(s(0) = 0\). Plug in t=0 into the position function to find the constant D: $$0 = -4.9(0)^2 + 20(0) + D \Rightarrow D = 0.$$ So, the position function is: $$s(t) = -4.9t^2 + 20t.$$ So, the position and velocity functions are: $$s(t) = -4.9t^2 + 20t$$ $$v(t) = -9.8t + 20$$

Key Concepts

Kinematics

Kinematics is the branch of physics that studies the motion of objects without considering the forces that cause the motion. It focuses on concepts such as velocity and acceleration to describe how an object's position changes over time. Understanding kinematics involves analyzing the paths and trajectories of objects and predicting future positions based on their current state of motion.

In kinematics, you often deal with one-dimensional motion, where objects travel along a straight path. This simplifies the calculation, as you only need to consider linear equations. By using kinematic equations and applying calculus, we can determine various aspects of motion, such as velocity and position, from given acceleration data.

For example, in the exercise provided, we use the given constant acceleration to calculate the velocity function of the object by performing integration. We then use this velocity function to find the position function, demonstrating the importance of kinematics in predicting motion.

Acceleration

Acceleration, represented by the symbol \(a(t)\), is a measure of how quickly an object's velocity changes over time. It is a vector quantity, meaning it has both magnitude and direction. In the given problem, the acceleration is constant, and its value is -9.8 m/s², which suggests the object is slowing down due to gravity.

When acceleration is constant, it simplifies the calculation of the velocity function because integration becomes straightforward. By integrating the acceleration function, we determine the velocity function and understand the effects of acceleration on motion.

• Negative acceleration indicates deceleration or slowing down.
• Constant acceleration makes calculations easier and allows straightforward integration to find velocity.

This fundamental understanding of acceleration is crucial for our analysis of one-dimensional motion, helping us predict how an object will move over time.

Velocity

Velocity is the rate at which an object's position changes with time. Unlike speed, which is a scalar quantity, velocity is a vector, meaning it has both a direction and magnitude. In the exercise, the velocity function is derived by integrating the acceleration, resulting in \(v(t) = -9.8t + 20\).

The initial velocity \(v(0) = 20\) m/s is essential to determining the constant in the velocity equation after integration. This initial condition helps us tailor the general velocity function to the specific scenario presented in the problem. Here are some key points:

• Velocity gives both speed and direction.
• It can be positive or negative, indicating direction relative to a chosen reference point.
• The initial velocity is crucial for accurately modeling motion.

Understanding velocity is central to predicting motion, as it allows us to track an object's progress along its path.

Position Function

The position function \(s(t)\) describes the location of an object at any given time, \(t\). It is derived by integrating the velocity function, giving us precise information about where an object is along its path over time. In the exercise, the position function is \(s(t) = -4.9t^2 + 20t\).

This function shows how the object's position changes considering its initial position \(s(0)\). Determining this involves making use of the initial condition given in the problem (where \(s(0) = 0\)) to solve for any constants arising from integration.

• Position function: Tells us the object's location over time.
• Direct result of integrating velocity.
• Incorporates initial conditions to fit the specific scenario.

The position function is vital because it gives us a full picture of the object's motion, allowing us to know where it is at any point in time based on known velocity and initial conditions.