Problem 44 - Essential Calculus Early Transcendentals

Question

Show that for motion in a straight line with constant acceleration \(a\) , initial velocity \(v_{0},\) and initial displacement \(s_{0},\) the displacement after time \(t\) is \(s=\frac{1}{2} a t^{2}+v_{0} t+s_{0}\)

Step-by-Step Solution

Answer

Displacement after time \(t\) is given by \(s = \frac{1}{2} a t^2 + v_0 t + s_0\).

Step 1: Understand the formula for displacement

To show the displacement, we must derive it using the variables provided: acceleration \(a\), initial velocity \(v_0\), and initial displacement \(s_0\). We aim to find the displacement \(s\) after time \(t\).

Step 2: Use the basic formula for displacement with constant acceleration

The formula that relates displacement \(s\), initial velocity \(v_0\), acceleration \(a\), and time \(t\) is given by:\[ s = v_0 t + \frac{1}{2}a t^2. \] Here, the first term \(v_0 t\) represents the displacement due to the initial velocity, and the second term \(\frac{1}{2}a t^2\) represents the displacement due to acceleration.

Step 3: Include initial displacement

To include the effect of any initial starting position, we add the initial displacement \(s_0\) to the equation. This gives us:\[ s = \frac{1}{2} a t^2 + v_0 t + s_0. \] This equation shows that displacement is the sum of the displacement from initial velocity, the displacement from acceleration over time, and the initial displacement.

Key Concepts

Displacement Formula

In physics, displacement refers to the change in position of an object. It is a vector quantity, which means it involves both magnitude and direction. When dealing with constant acceleration, the displacement formula helps us calculate the position of an object after a certain period of time. The formula for displacement, when considering initial velocity and constant acceleration, is:

• \( s = v_{0}t + \frac{1}{2} a t^2 + s_0 \)

This expression allows us to find the total change in position by summing up the contributions from initial velocity, acceleration, and any initial starting position. The formula gives a clear depiction of how motion parameters influence position.

Initial Velocity

Initial velocity is a crucial concept in the study of motion. It represents the speed and direction of an object at the beginning of the time period considered. When an object starts moving, \( v_{0} \) reflects its starting speed.
In the displacement formula, the initial velocity is a vital component. It contributes to the term \( v_{0}t \), indicating the distance an object would cover due to its starting speed alone, excluding any effect of acceleration.
Understanding initial velocity helps us conceptualize how fast and in which direction an object begins its journey.

Time-Dependent Displacement

Time-dependent displacement analyzes how an object's position changes over time. As time \( t \) progresses, the displacement grows as a function of time, influenced by both initial conditions and the object's acceleration.
The displacement formula is very reliant on time because it's squared in the acceleration term, \( \frac{1}{2}a t^2 \). This quadratic relationship shows that displacement increases with the square of time, representing an ever-accelerating motion.

• If \( a = 0 \), displacement becomes linear, governed solely by initial velocity.
• When \( a > 0 \), the displacement grows rapidly due to the increasing speed over time.

Considering time-dependent displacement allows us to predict future positions based on current motion conditions.

Kinematics Equation

The kinematics equation for displacement is integral to understanding motion. Kinematics is the branch of physics that studies movement without considering the forces that cause it. This equation:
\[ s = \frac{1}{2}a t^2 + v_{0}t + s_0 \]
is a classic example of how physics models motion scenarios using just a few variables.

• \(\frac{1}{2} a t^2 \) expresses the impact of constant acceleration stretching over time.
• \(v_{0}t \) highlights the influence of the object's initial speed on its journey.
• \(s_0\) involves any initial displacement, setting where the object starts.

These components together allow us to describe the complete trajectory of an object's motion, providing insights into past, present, and future positions.