Instantaneous Velocity, Acceleration, Jerk, Slopes, Graphs vs. Time | Doc Physics

Summary

This educational video script explores the concepts of velocity and acceleration in physics through graphical representations. It explains how a straight line in a position-time graph indicates constant velocity, while varying slopes suggest changes in speed. The script delves into calculating average velocity, introduces the idea of instantaneous velocity as a limit, and discusses acceleration as the rate of change of velocity. It also humorously introduces 'jerk' as the rate of change of acceleration, relating it to the sensation of balance shifts in everyday scenarios like braking in a car.

Takeaways

• 📈 A straight line in a position versus time graph indicates constant velocity, where the slope represents the rate of change of position over time.
• 🐧 The concept of velocity being the slope of a position-time graph is illustrated using a penguin as an example, emphasizing the physical meaning of the slope.
• 📚 The script suggests watching calculus videos for a deeper understanding of the relationship between calculus and physics, highlighting their interconnectedness.
• 🔍 The constant slope in a velocity-time graph is explained as the marker always being tangent to the line, indicating uniform motion.
• 🤔 The script introduces the idea of calculating velocity between specific time intervals using the rise over run method, which approximates the instantaneous velocity.
• 🎯 The concept of a limit is discussed, defining it as the process of narrowing the time interval to find the instantaneous velocity as delta T approaches zero.
• 📉 The script describes a scenario where the velocity graph is not symmetric, showing how velocity changes over time and eventually becomes negative.
• 🔄 The rate of change of velocity is introduced as acceleration, which is the slope of the velocity-time graph, measured in meters per second squared.
• 🚀 The script proposes the concept of 'jerk' as the rate of change of acceleration, with units of meters per second cubed, relating it to the physical sensation of sudden changes in motion.
• 📊 A table summarizing the units of position, displacement, velocity, acceleration, and jerk is suggested to highlight the pattern of these measurements being divided by time.
• 👋 The video concludes with a teaser for another video, indicating that further explanation and discussion will be provided in a follow-up.

Q & A

What is the physical meaning of the slope in a position versus time graph?

The slope in a position versus time graph represents the velocity of an object. It is calculated as the change in position (rise) over the change in time (run), which is mathematically expressed as \( V_{average} = \frac{\Delta x}{\Delta t} \).

Why is calculus important in understanding the physics of motion?

Calculus is important because it was invented to help with problems in physics, such as determining instantaneous velocity and acceleration. It allows for the analysis of rates of change and the behavior of objects at specific points in time.

What does a constant slope in a velocity versus time graph indicate about an object's motion?

A constant slope in a velocity versus time graph indicates that the object is moving with a constant velocity, meaning there is no acceleration or deceleration occurring.

How can you determine the instantaneous velocity of an object at a specific time?

The instantaneous velocity can be determined by taking the limit of the change in position over the change in time as the time interval approaches zero (\( \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} \)).

What is the concept of acceleration in physics?

Acceleration is the rate of change of velocity with respect to time. It is calculated as the change in velocity (delta V) over the change in time (delta T), or \( a = \frac{\Delta v}{\Delta t} \).

How does the graph of velocity as a function of time help in understanding an object's motion?

The graph of velocity versus time shows how the velocity of an object changes over time. It can indicate whether the object is speeding up, slowing down, or moving at a constant speed.

What does a negative velocity indicate in the context of the graph?

A negative velocity indicates that the object is moving in the opposite direction to the positive reference direction defined in the graph.

What is the term used for the rate of change of acceleration?

The rate of change of acceleration is referred to as 'jerk,' which is calculated as the change in acceleration (delta a) over the change in time (delta T), or \( j = \frac{\Delta a}{\Delta t} \).

How does the concept of a limit relate to finding instantaneous velocity?

The concept of a limit is used to find the instantaneous velocity by narrowing the time interval (\( \Delta t \)) to an infinitely small value, thus providing the velocity at a specific instant in time.

Why is the slope of the acceleration graph important?

The slope of the acceleration graph is important because it represents the rate at which the velocity of an object is changing. A positive slope indicates increasing velocity, while a negative slope indicates decreasing velocity.

What units are used to measure acceleration?

Acceleration is measured in meters per second squared (\( m/s^2 \)) because it involves the change in velocity (meters per second) over time (seconds).

Outlines

00:00

📚 Understanding Velocity and Instantaneous Velocity

The script begins with an exploration of velocity, emphasizing that a straight line on a position versus time graph indicates constant velocity, which is the slope of the line. It explains the concept of average velocity as the change in position (delta X) over the change in time (delta T). The speaker then introduces the idea of instantaneous velocity, which is found by taking the limit of delta X over delta T as the time interval approaches zero. This concept is illustrated with a graph of a penguin's motion, showing how to calculate the velocity at different points in time by creating similar right triangles on the graph.

05:02

🚀 Introducing Acceleration and Jerk

The second paragraph delves into acceleration, which is the rate of change of velocity over time. The speaker uses a graph to demonstrate how the slope of the velocity versus time graph represents acceleration. The graph shows a negative acceleration initially, which increases in magnitude until it reaches zero, indicating a change in the motion of an object. The concept of jerk is introduced as the rate of change of acceleration, with units of meters per second cubed, and is likened to the feeling of a sudden shift in balance, such as when a car brakes or a subway starts or stops. The script concludes with a brief mention of creating a table to summarize the concepts of position, displacement, velocity, acceleration, and jerk, and hints at further discussion in a subsequent video.

Keywords

💡Position

Position refers to the location of an object in space. In the context of the video, it is a fundamental concept in kinematics, the branch of physics that deals with the motion of objects. The script discusses how a straight line in a position versus time graph indicates a constant velocity, which is the rate of change of position with respect to time. The video uses the example of a penguin moving in a straight line to illustrate this concept.

💡Velocity

Velocity is a vector quantity that represents the rate of change of an object's position with respect to time. The script explains that velocity can be determined from a velocity versus time graph, where a constant slope indicates a constant velocity. The video also introduces the concept of instantaneous velocity, which is the velocity at a specific moment in time, as opposed to average velocity over a period of time.

💡Slope

Slope, in the context of the video, is the measure of the steepness of a line on a graph. It is a key concept when discussing graphs of position and velocity versus time. The video script uses the slope of the line to explain how to calculate the velocity of an object, as the slope represents the change in position (rise) over the change in time (run).

💡Graph

A graph is a visual representation of data, often used in physics to represent the relationship between two variables. In the video, graphs are used to illustrate the concepts of position, velocity, and acceleration over time. The script discusses how different shapes of graphs can represent different types of motion, such as constant velocity or acceleration.

💡Calculus

Calculus is a branch of mathematics that deals with rates of change and accumulation. The script mentions calculus in the context of understanding the physics of motion, particularly when discussing instantaneous velocity and acceleration. The video suggests that calculus is essential for analyzing the rate of change of velocity and position over time.

💡Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific instant in time, as opposed to average velocity. The video script explains that it can be found by taking the limit of the change in position over the change in time as the time interval approaches zero. This concept is crucial for understanding the precise motion of an object at any given moment.

💡Acceleration

Acceleration is the rate of change of velocity with respect to time. The video script introduces acceleration as the slope of the velocity versus time graph, indicating how the velocity of an object changes over time. The concept is used to describe scenarios where the velocity of an object increases or decreases, such as when the penguin in the video comes to a stop and then starts moving in the opposite direction.

💡Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The script briefly mentions trigonometry when discussing the formation of right triangles on a graph to determine the slope, which is essential for calculating velocity and acceleration.

💡Limit

In mathematics, a limit is the value that a function or sequence approaches as the input approaches some value. The video script uses the concept of a limit to describe how instantaneous velocity is found by considering the velocity over an infinitesimally small time interval, as the time interval approaches zero.

💡Jerk

Jerk, in physics, is the rate of change of acceleration with respect to time, and it is measured in meters per second cubed. The script introduces jerk as a concept that describes the sudden change in the rate of motion, such as the feeling of being pushed back into your seat when a car accelerates or the sudden shift when a subway car starts or stops.

Highlights

The physical meaning of the slope in a position versus time graph is explained as the change in position (rise) over the change in time (run), resulting in average velocity.

A constant slope in a velocity versus time graph indicates uniform motion, with the slope representing the constant velocity.

The use of calculus in physics to understand the relationship between velocity and time, especially when the motion is not uniform.

The concept of a limit in calculus is introduced to find the instantaneous velocity by narrowing the time interval (delta T) to zero.

Instantaneous velocity is defined as the velocity at a specific moment, derived from the limit of the ratio of change in position to change in time as time approaches zero.

The importance of recognizing the difference between average velocity and instantaneous velocity, especially in non-uniform motion.

The graphical representation of velocity as a function of time for a non-uniformly moving object, showing changes in the slope that indicate changes in velocity.

The introduction of acceleration as the rate of change of velocity, analogous to how velocity is the rate of change of position.

Acceleration is calculated as the change in velocity (delta V) over the change in time (delta T), with units of meters per second squared.

The graphical interpretation of acceleration as the slope of the velocity versus time graph, indicating how velocity changes over time.

The concept of jerk, or the rate of change of acceleration, introduced with units of meters per second cubed, relating to the feeling of sudden changes in motion.

Jerk is described as the sensation of balance shift during sudden starts or stops in motion, such as in a car or subway.

The significance of understanding the patterns of motion through the analysis of position, velocity, acceleration, and jerk.

The educational value of the video in demystifying the concepts of calculus as they apply to physics and motion.

The practical applications of these concepts in understanding and predicting the behavior of objects in motion.

The encouragement for viewers to explore further calculus videos for a deeper understanding of the physics of motion.

Transcripts

Hi, we saw in the previous video that

if a position versus time graph is a straight line

then we've got a constant slope and that slope

has physical meaning because the slope well we can take rise and run we can go dot

dot dot dot dot dot dot

dot dot dot dot dot dot dot make a little right triangle right there and we can identify

the rise as the change in

X and we can identify the run

as the change in time and we found that V

average is delta X over delta T. In this case though

I mean it's not just V average it' s just V for this poor little penguin

chugging along right here

but what might be interesting is if we have a graph that looks

a little bit more interesting if we look at a graph like

this. Oh, let me talk about what this the velocity of this thing is positive

right?

This poor penguin here so we'll get a graph down here and we could, for the same

experience, we could do a graph of velocity versus time.

You should watch some calculus videos I have a few

if you're interested, videos that will help you understand

the calculus of physics because calculus was invented

for Newton to do his physics. I guess Leibnitz was around too. But

look at this, the slope of this thing is always the same we can do that by

putting a marker on here and seeing that

the marker always is tangent to the line.

See, it's always tangent to the line and always has the same slope. So that means

that the slope is constant and velocity

is the slope, so I can make a little graph

of the velocity as a function of time and I find it to always be the same.

And whatever that value is is the calculated slope that we would get from

analyzing any segment along here you see how all of them would give the same

slope. If I chose this point

and this point I'd make another little right triangle but it's a similar

triangle right it has all the same angles

same ratio of rise over run.

Oh boy! That's smacks of trigonometry, lovely.

But what if we've got something more interesting happening? What if we've got

something like

this happening with our position as a function of time?

I deliberately didn't make it symmetric, that's not a mistake right there.

So let's say I want to know the velocity between let's say one second

and four seconds. Uh one second,

four seconds, I guess I can put two and three in here as well.

But in between one second and four seconds

I can do a little bit of rise and run, there's the run

there's the rise and if I connect those guys I could find the slope

of that line right there. Now you wouldn't say,

necessarily, that that's the same as the velocity at this moment right here 2.5

seconds I might be interested in that exact velocity though

but the thing is I'd have to zoom in a little bit further

now I might take the, uh, take the velocity between

two seconds and three seconds, so if I go between here

and here I'd find a line that's similar slope but probably not exactly the same.

But ultimately I want to narrow in and more and more and more and more. So what we're

doing is called

a limit, and I'm gonna define the limit

as T approaches zero of

delta X over delta T as I zoom in smaller and smaller and smaller right

here

I'm going to get the instantaneous velocity I'm just gonna call that V 'cause

I can.

This is instantaneous, meaning

that its the velocity at a particular instant.

So watch the calculus videos you'll see this make a lot more sense.

But if I were to make a graph of velocity as a function of time

for that guy, let's try that a little bit. I'm gonna take

this guy and analyze its velocity as a function of time.

Mmmhmm, here's what I'm thinking,

I'm thinking initially the velocity is large. Look at the slope here, it's very large

velocity initially and then the velocity get smaller and smaller and smaller until

it eventually hits 0

in fact it hits 0 at right about three seconds.

I'll put a tick here at three seconds and say that the velocity has to have hit zero

somehow, it'd be like

somehow hitting zero and then, oh dang, look the velocity

is the slope right. So the velocity is negative here and it becomes

steady so what I gotta do is I gotta kinda curve out here

and say the velocity becomes steady because here the slope seems to be

constantly

low constantly negative. So this velocity graph of that guy's doing some funky like

that.

That was my deliberate not making it, uh,

symmetric argument right there. So

it's interesting though since velocity changes sometimes, remember this graph here

of velocity, we could actually interest ourselves in the slope of this graph.

So to that and I'll propose to you a new concept

that by analogy, remember we had delta X

over delta T was average velocity by analogy to that I'm just gonna take the

same dang thing

and say delta V over delta T.

That might be interesting right? And what should we call it? Well it'll be

something average just exactly the same format.

Turns out that that concept is called acceleration and the acceleration talks

about the rate

of change, that means divided by T, the rate of

change of velocity. So if we look at that we'll be thinking about the slope

of this graph *he's making that ch ch ch noise again*

as a function of time so I need to put that below us again

and I'm gonna look at this graph right here, I like to keep this format where I'm

always looking below

and now I'm going to graph the acceleration of this weird

thing as a function of time and I see

that the slope here, slope

is acceleration. This is actually a really big deal that the slope is

acceleration right here

the slope is acceleration and the acceleration starts out negative

and then the slope gets bigger and bigger and bigger and eventually reaches

zero.

So I haven't left myself much room I'm gonna have to scoot up

a tiny bit, see if you guys can still rock and see everything on screen.

Here's the plan, I'm starting negative and constant,

and then it gets bigger and bigger here and eventually reaches

0 it looks like there is no slope

in this region, and that corresponds here,

and the slope is negative right there and that corresponds that time right there. See it's

kind of straight

and then something weird happens right here and we get to zero. I'm just trying to give

you an idea

about acceleration. So let's make a little table of some stuff that we

know.

We know three things now. I guess we know a lot about time right,

but we'll just talk about the distance related things, we know

position and displacement that's all measured in meters

we know velocity and that's measured in meters per second 'cause velocity

is , well we have delta X over delta T,maybe we're taking a limit or not but,

we're dividing by T.

And then we get delta V

over delta T which means we're dividing by time again so I'm gonna say

acceleration

has units have meters per second squared, oh dang.

Do you think we keep going? This is very clearly a pattern of dividing by T.

What if I say then delta acceleration divided by delta T well that would have

units at meters per second cubed

I'm gonna call that sucker jerk, this is a real scientific term.

It's the feeling that you get like a shifting of your balance

when you're breaking a car, somebody else being a car probably, 'cause then you don't

anticipate it, right.

The car begins to break and suddenly your balance has shifted, or when a

subway car starts or stops think about that,

think about that feeling you get. That is called jerk.

Okay so now I want to summarize what's happening if you've got,

no, let's do it in another video. Bye-bye.