how to find instantaneous velocity from a graph

3.3: Instantaneous Velocity and Speed

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Learning Objectives

• Explain the divergence between average velocity and instantaneous velocity.
• Describe the difference betwixt velocity and speed.
• Calculate the instantaneous velocity given the mathematical equation for the velocity.
• Summate the speed given the instantaneous velocity.

We take now seen how to summate the average velocity between ii positions. However, since objects in the real earth motion continuously through space and time, we would like to detect the velocity of an object at whatsoever single betoken. Nosotros can find the velocity of the object anywhere along its path by using some central principles of calculus. This department gives u.s. improve insight into the physics of motion and volition exist useful in later chapters.

Instantaneous Velocity

The quantity that tells usa how fast an object is moving anywhere along its path is the instantaneous velocity, unremarkably called simply velocity. Information technology is the average velocity betwixt two points on the path in the limit that the time (and therefore the deportation) between the two points approaches naught. To illustrate this idea mathematically, we need to limited position x as a continuous function of t denoted by x(t). The expression for the boilerplate velocity between two points using this annotation is \(\bar{5} = \frac{x(t_{2}) - x(t_{one})}{t_{two} - t_{1}}\). To find the instantaneous velocity at any position, nosotros let t₁ = t and t₂ = t + \(\Delta\)t. Later inserting these expressions into the equation for the average velocity and taking the limit as \(\Delta\)t → 0, nosotros find the expression for the instantaneous velocity:

\[5(t) = \lim_{\Delta t \to 0} \frac{ten(t + \Delta t) - x(t)}{\Delta t} = \frac{dx(t)}{dt} \ldotp\]

Instantaneous Velocity

The instantaneous velocity of an object is the limit of the average velocity as the elapsed fourth dimension approaches zero, or the derivative of x with respect to t:

\[v(t) = \frac{d}{dt} x(t) \ldotp \label{three.iv}\]

Like boilerplate velocity, instantaneous velocity is a vector with dimension of length per time. The instantaneous velocity at a specific time point t₀ is the rate of change of the position function, which is the slope of the position function ten(t) at t₀. Figure \(\PageIndex{i}\) shows how the boilerplate velocity \(\bar{v} = \frac{\Delta x}{\Delta t}\) between two times approaches the instantaneous velocity at t₀. The instantaneous velocity is shown at time t₀, which happens to exist at the maximum of the position function. The gradient of the position graph is null at this point, and thus the instantaneous velocity is zip. At other times, t_i, t₂, then on, the instantaneous velocity is not zero because the slope of the position graph would be positive or negative. If the position function had a minimum, the slope of the position graph would also be zip, giving an instantaneous velocity of zero there as well. Thus, the zeros of the velocity function requite the minimum and maximum of the position function.

Figure \(\PageIndex{1}\): In a graph of position versus time, the instantaneous velocity is the slope of the tangent line at a given point. The average velocities \(\bar{5} = \frac{\Delta x}{\Delta t} = \frac{x_{f} - x_{i}}{t_{f} - t_{i}}\) between times \(\Delta\)t = t₆ − t₁, \(\Delta\)t = t₅ − t₂, and \(\Delta\)t = t_iv − t₃ are shown. When \(\Delta\)t → 0, the average velocity approaches the instantaneous velocity at t = t₀.

Example 3.2: Finding Velocity from a Position-Versus-Time Graph

Given the position-versus-time graph of Figure \(\PageIndex{2}\), find the velocity-versus-time graph.

Figure \(\PageIndex{two}\): The object starts out in the positive direction, stops for a short time, and then reverses management, heading back toward the origin. Notice that the object comes to rest instantaneously, which would require an space forcefulness. Thus, the graph is an approximation of motion in the existent world. (The concept of force is discussed in Newton's Laws of Motion.)

Strategy

The graph contains three directly lines during 3 fourth dimension intervals. We find the velocity during each time interval by taking the slope of the line using the grid.

Solution

Time interval 0 s to 0.5 s: \(\bar{v} = \frac{\Delta x}{\Delta t}=\frac{0.5\; m − 0.0\; thousand}{0.v\; south − 0.0\; s} = one.0\; g/s\)

Time interval 0.5 s to 1.0 due south: \(\bar{5} = \frac{\Delta x}{\Delta t}=\frac{0.0\; 1000 − 0.0\; g}{1.0\; s − 0.5\; s} = 0.0\; thou/s\)

Time interval 1.0 s to 2.0 s: \(\bar{5} = \frac{\Delta ten}{\Delta t}=\frac{0.0\; grand − 0.5\; m}{two.0\; s − i.0\; s} = -0.5\; m/s\)

The graph of these values of velocity versus time is shown in Figure \(\PageIndex{3}\).

Effigy \(\PageIndex{iii}\): The velocity is positive for the start office of the trip, zilch when the object is stopped, and negative when the object reverses direction.

Significance

During the fourth dimension interval between 0 south and 0.5 s, the object's position is moving abroad from the origin and the position-versus-time curve has a positive slope. At whatsoever point along the bend during this time interval, we can find the instantaneous velocity by taking its gradient, which is +1 m/south, equally shown in Figure \(\PageIndex{three}\). In the subsequent time interval, between 0.5 s and 1.0 s, the position doesn't alter and we run across the slope is zero. From 1.0 s to 2.0 s, the object is moving dorsum toward the origin and the slope is −0.5 chiliad/s. The object has reversed direction and has a negative velocity.

Speed

In everyday linguistic communication, most people use the terms speed and velocity interchangeably. In physics, all the same, they practice non have the aforementioned meaning and are singled-out concepts. I major difference is that speed has no direction; that is, speed is a scalar.

Nosotros can calculate the average speed by finding the total distance traveled divided by the elapsed fourth dimension:

\[Boilerplate\; speed = \bar{s} = \frac{Total\; distance}{Elapsed\; time} \ldotp \label{3.5}\]

Average speed is non necessarily the aforementioned equally the magnitude of the boilerplate velocity, which is plant by dividing the magnitude of the total displacement by the elapsed fourth dimension. For example, if a trip starts and ends at the same location, the full deportation is cypher, and therefore the average velocity is zero. The boilerplate speed, nevertheless, is not zero, because the total distance traveled is greater than zero. If nosotros take a route trip of 300 km and need to be at our destination at a certain fourth dimension, then we would be interested in our average speed.

Even so, we tin calculate the instantaneous speed from the magnitude of the instantaneous velocity:

\[Instantaneous\; speed = |v(t)| \ldotp \label{iii.6}\]

If a particle is moving along the x-axis at +seven.0 m/s and another particle is moving along the same centrality at −vii.0 k/s, they have different velocities, simply both have the same speed of 7.0 m/s. Some typical speeds are shown in the post-obit table.

Table iii.1 - Speeds of Various Objects

Speed	m/south	mi/h
Continental drift	ten-7	2 x 10-7
Brisk walk	1.7	3.nine
Cyclist	4.4	10
Sprint runner	12.2	27
Rural speed limit	24.6	56
Official land speed record	341.ane	763
Speed of audio at sea level	343	768
Space shuttle on reetry	7800	17,500
Escape velocity of Earth*	11,200	25,000
Orbital speed of Earth around the Sunday	29,783	66,623
Speed of light in a vacuum	299,792,458	670,616,629
*Escape velocity is the velocity at which an object must be launched and so that information technology overcomes Earth's gravity and is not pulled back toward Globe.

Calculating Instantaneous Velocity

When computing instantaneous velocity, we need to specify the explicit course of the position function x(t). If each term in the x(t) equation has the form of Atⁿ where A is a abiding and n is an integer, this can be differentiated using the power dominion to be:

\[\frac{d\left(A t^{north}\correct)}{d t}=A n t^{n-1} \ldotp \label{3.seven}\]

Notation that if in that location are additional terms added together, this power rule of differentiation can be done multiple times and the solution is the sum of those terms. The post-obit example illustrates the employ of Equation \ref{3.7}.

Example iii.iii: Instantaneous Velocity Versus Average Velocity

The position of a particle is given by x(t) = 3.0t + 0.5t³ k.

1. Using Equation \ref{3.four} and Equation \ref{iii.7}, observe the instantaneous velocity at t = 2.0 southward.
2. Calculate the boilerplate velocity betwixt 1.0 s and iii.0 due south.

Strategy

Equation \ref{3.four} give the instantaneous velocity of the particle equally the derivative of the position function. Looking at the form of the position function given, we see that it is a polynomial in t. Therefore, we tin apply Equation \ref{3.7}, the ability rule from calculus, to discover the solution. We utilise Equation \ref{three.half-dozen} to calculate the average velocity of the particle.

Solution

1. v(t) = \(\frac{dx(t)}{dt}\) = iii.0 + 1.5t² thou/s. Substituting t = two.0 s into this equation gives v(two.0 south) = [iii.0 + 1.5(ii.0)ⁱⁱ] m/south = 9.0 thou/s.
2. To decide the average velocity of the particle between 1.0 s and 3.0 s, we calculate the values of x(ane.0 s) and x(3.0 due south):

\[x(1.0 s) = \large[(3.0)(i.0) + 0.5(1.0)^{3} \big]m = 3.5\; m\]

\[ten(iii.0 s) =\big[(3.0)(3.0) + 0.5(3.0)^{three}\big] m = 22.5\; one thousand\]

Then the average velocity is

\[\bar{v} = \frac{x(three.0\; s) - x(ane.0\; southward)}{t(3.0\; south) - t(1.0\; south)} = \frac{22.5 - 3.5\; thou}{three.0 - ane.0\; due south} = 9.five\; chiliad/s \ldotp\]

Significance

In the limit that the time interval used to calculate \(\bar{5}\) goes to zero, the value obtained for \(\bar{five}\) converges to the value of v.

Example 3.4: Instantaneous Velocity Versus Speed

Consider the motion of a particle in which the position is 10(t) = 3.0t − 3t² m.

1. What is the instantaneous velocity at t = 0.25 s, t = 0.fifty s, and t = one.0 s?
2. What is the speed of the particle at these times?

Strategy

The instantaneous velocity is the derivative of the position office and the speed is the magnitude of the instantaneous velocity. We use Equation \ref{3.4} and Equation \ref{3.7} to solve for instantaneous velocity.

Solution

1. v(t) = \(\frac{dx(t)}{dt}\) = 3.0 − 6.0t g/southward
2. v(0.25 s) = ane.fifty m/south, v(0.v south) = 0 yard/s, five(1.0 s) = −3.0 m/due south
3. Speed = |v(t)| = 1.50 m/s, 0.0 thou/s, and 3.0 1000/s

Significance

The velocity of the particle gives u.s.a. management information, indicating the particle is moving to the left (west) or right (east). The speed gives the magnitude of the velocity. By graphing the position, velocity, and speed as functions of fourth dimension, nosotros can understand these concepts visually Effigy \(\PageIndex{4}\). In (a), the graph shows the particle moving in the positive direction until t = 0.5 s, when it reverses direction. The reversal of management can likewise be seen in (b) at 0.v southward where the velocity is cypher and then turns negative. At one.0 due south it is back at the origin where it started. The particle's velocity at 1.0 southward in (b) is negative, because it is traveling in the negative direction. But in (c), all the same, its speed is positive and remains positive throughout the travel time. We tin can too translate velocity as the slope of the position-versus-time graph. The slope of ten(t) is decreasing toward cypher, becoming zero at 0.5 s and increasingly negative thereafter. This analysis of comparing the graphs of position, velocity, and speed helps catch errors in calculations. The graphs must be consistent with each other and help interpret the calculations.

Figure \(\PageIndex{4}\): (a) Position: ten(t) versus time. (b) Velocity: 5(t) versus fourth dimension. The gradient of the position graph is the velocity. A crude comparing of the slopes of the tangent lines in (a) at 0.25 s, 0.5 s, and 1.0 southward with the values for velocity at the respective times indicates they are the same values. (c) Speed: |v(t)| versus time. Speed is e'er a positive number.

Exercise iii.two

The position of an object as a function of time is x(t) = −3t² thousand. (a) What is the velocity of the object equally a function of time? (b) Is the velocity ever positive? (c) What are the velocity and speed at t = 1.0 s?

Source: https://phys.libretexts.org/Bookshelves/University_Physics/Book:_University_Physics_%28OpenStax%29/Book:_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_%28OpenStax%29/03:_Motion_Along_a_Straight_Line/3.03:_Instantaneous_Velocity_and_Speed

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