Motion

Name the metric units of distance, time, and speed and give their symbols.
Define the term speed.
Know which simple instruments can be used to measure distance and time.
Identify the relationship between distance, speed and time and use the formula in calculations.
Solve problems involving distance, speed and time when data is given in non-standard units.
Use the concepts of instantaneous speed, average speed and constant speed to answer questions.

Velocity = Distance / Time

Velocity is the same as Speed most of the time

In some physics Speed is how fast you are traveling.

E.g. 5 Km/h

Velocity is Speed in a direction.

E.g. 5 Km/h North

Velocity defined and as an equation

Velocity is the rate at which something moves.

It can change, it can also be constant.

When you are in a car it can change from 0 km/h.

It can then be a constant 100 km/h

Notice how 100 km/h is written. The km is first.

You say kilometers per hour.

The distance is before the time

Like distance/time

or d/t

so d/t = v

Velocity = Distance / time

v = d/t

So, Velocity is the rate at which an object covers a distance in a specified unit of time

Rearranging formula: Simple Mathematics or Triangles

Simple maths-ing

v = d/t

How do I isolate 'd'

do simple maths on it

v = d/t becomes 3 = 6/2

So now v = 3, d = 6 and t = 2.

So if we want to isolate 'd' which we have changed to 6

Then how can we rearrange the numbers 3, 6 and 2 so that the answer is 6

6 = 3 x 2

d = v*t

Triangle-ing

Cover the 'd' in the triangle

Note you are not given the triangles

The calculation for Speed and for Velocity is exactly the same

How fast are you going?

Well measure a distance

For instance Usain Bolt and James Corden are going to run 100 meters

So they run the same distance

The difference between them is how much time it takes them to run that distance

It takes Usain Bolt 11 seconds

It takes James Corden 15 seconds

So who was faster? - Usain Bolt

How fast was he? What was his Velocity?

They both ran 100 meters

Usain Bolt ran 100 meters in 11 seconds

James Corden ran 100 meters in 15 seconds

How many meters could they travel in each second? To do this we would take the distance they traveled and divide it by time. This would give the distance traveled (meters) in each unit of time (seconds)

Usain Bolt:

100 meters / 11 seconds = 9 meters per second

James Corden:

100 meters / 15 seconds = 6 meters per second

This shows that in each second, Usain Bolt covered more distance. So he was faster

The bigger the Velocity number the greater the distance covered per unit of time, so the faster they are going

If we look again at the maths, we see that we did:

meters / seconds = meters per second

This is the same on a Car's Speedometer:

kilometers / hour = kilometers per hour

What do these units represent?

distance / time = speed

Because speed and velocity are basically the same in most settings, we can say

distance / time = velocity

This is saying that velocity is distance divided by time

velocity = distance / time

We can also just use letters

v = d/t

So we can see that Velocity = distance / time

But what if we don't want to find velocity, rather we want to find distance or time?

For instance, I'm on the motorway at Pokeno and I want to head north to Whangarei. The Distance of road between the two points is 200km

I'm driving at 100km/h. Lets assume there is no traffic issues, no intersections etc so I can set the cruise control to 100km per hour

How much time will it take to travel the 200km?

Well, you would have worked that out in your head.

Traveling at 100kmph it would take 2 hours to travel 200km

What does that look like if you show your working?

I'd go:

200km / 100kmph = 2 hours

So what did I do? I went:

distance / velocity = time

What if I want to find distance?

Say my you borrow my car for 3 hours and drive the entire time. My car can tell me the average speed, it says that you drove it at 50km per h

So, how far did you travel?

Again, you kind of know how to calculate that without ever being told how to. You probably calculated it as 150km. But how did you do that? What was your working?

I'd go:

50kmph x 3 hours = 150km

So what did I do? I went:

velocity x time = distance

What we have done is called 'rearranging formula'

I personally think it is mostly intuitive, sometimes with hard questions, I'll insert simple maths first, like 6 = 3/2 and then rearrange that to find out how to rearrange the formula and then use the rearranged formula to work out the question.

You can also memorize triangles and use those to help with the rearranging. Below is the triangle. And to the side is a clip showing you how to do that. But I don't do it. :-)

Velocity or Speed

Is it speed or is it velocity?

Speed and Velocity use the same formula

But there is a difference

Speed is in any direction, it is what your speedometer in your car tells you.

Whereas Velocity is in a particular direction

So, if you were grounded and your parents checked on you at the start and end of each hour, the hour might play out as follows. They check and you are in your room, Then you snuck out to your friends place next door for 30 minutes, then you quickly snuck home again. And then your parents checked again an you were in your room. To your parents your Velocity was 0 km per hour

This is because you left your start point and then returned to your start point, so ultimately you went nowhere. Your displacement (change in location) is zero.

You could actually go a step further. As velocity is determined by displacement over time, and you choose what that time value is.

So, you wake up in the morning in your bed. You go to school, or drive past it due to lockdown, then return home, then go to sleep in your bed. So your displacement for the day is zero. And thus your velocity is zero.

However, your speed over the course of the day changes throughout the day, and you could calculate an average (by adding up the distance covered in total and dividing it by the unit of time that you have chosen) speed.

The reason they are different is Speed is motion in ANY direction as it is Scalar, and only takes into consideration the magnitude. But velocity needs direction as it is a vector (has an arrow).

Watch the GPB Video "what are speed and Velocity" as it is very good

Also, Y11 all of the velocities are in one direction

Instantaneous Velocity vs Constant Velocity vs Average Velocity

Average Velocity

Average Velocity is the velocity that an object takes over a given unit of time

You might walk to McDonalds. Which is 1800 meters North of your house, taking 15 minutes, grab your takeaways, then return home, again taking 15 minutes.

Because your displacement over an hour is 0 meters, as you returned to your start point, your average velocity is 0 meters per hour

Average Speed

Speed doesn't depend on direction, so average speed is the average rate of motion in any direction. Because of this, with the McDonald's example we can take the total distance of 1800 x 2 = 3600 meters and the total time 15 x 2 = 30 minutes x 60 seconds = 1800 seconds

s = d/t

s = 3600m / 1800s

s = 2m/s

Instantaneous Velocity

Instantaneous Velocity is the rate of displacement that the object would travel if shortened to fit within the unit of time.

So, referring to the journey to McDonalds. You are walking their, but daydreaming, so you are walking slowly, then you see a person that you don't like but they haven't seen you, so you briefly run at a velocity of 10m/s for 20 seconds to increase the gap between you and that annoying person.

If we take two time points, 3 seconds before your run and 3 seconds into your run, you may find that in the first 'instant' you were travelling with a velocity of 1m/s, whilst at the second instant you were traveling with a velocity of 10m/s. These two velocities at these two instances are two examples of instantaneous velocity - the velocity of an object in any given instant

Constant Velocity

On your journey to McDonalds, you see a dollar on the ground. You slow down to check out a flash car. You speed up to avoid talking to that annoying person. Your Instantaneous Velocity fluctuates. So, if you were to pick any moment of the journey on the instantaneous velocity may have different readings

However, on your Journey back you needed to pee, so you walked at a constant pace. This is Constant Velocity, as you can pick any moment of it and the velocity will be the same: 2 m per second South

Lets watch the clip. We will record the velocity at 3 random moments in time. These instants will be at 10 seconds, 20 seconds and 43 seconds

At 10 seconds we can see that in that instant the velocity shown on the speedometer is: 56km/h

At 20 seconds we can see that in that instant the velocity shown on the speedometer is: 87km/h

At 43 seconds we can see that in that random instant the velocity shown on the speedometer is: 102km/h

Each of these readings gives us the speedometer readings at that moment in time, in that instant. The velocity in that instant is called Instantaneous Velocity.

Instantaneous Velocity = velocity in that instant

Your speedometer tells you your velocity in that moment, thus your speedometer tells you your Instantaneous Velocity

In this drive the car travels a distance of 900 meters in 52 seconds

We can work out the average velocity by:

velocity = distance / time

velocity = 900m / 52s

velocity = 17 m/s

So on average the car travels a distance of 17 meters in every second

out of curiosity we can convert this to kilometers per hour by timesing it, or multiplying it by 3.6.

17m/s x 3.6 = 62km/h

This gives us 62km/h

So the Average Velocity of the car is 62km/h

On the motorway, I might decide to use my cruise control. This will keep the car cruising at a constant velocity of 100km/h

Thus the Constant Velocity = 100km/h

When using cruse control you are trying to get the car to maintain a constant velocity. This is helpful in places where you don't need to speed up or slow down too often - such as on the motorway when there isn't much traffic

Constant Velocity is when the velocity doesn't change

Motorway Journey

Olivia is on the motorway, she'll drive the 10km from Newmarket Viaduct to Otahuhu

For the first 5 minutes her speedometer reads - 100km per hour - For the first 5 minutes her velocity is constant and the speedometer reading is of Instantaneous Velocity

During these 5 minutes she will travel a distance of:

v = d/t (rearrange using simple maths to isolate d, you can also use the triangle.... v = d/t... 3 = 6/2 ... 6 = 3x2 .. so ...d = v*t
d = 100km/h x ((1 hour / 60 minutes) * 5 minutes)
d = 100km/h x 0.083 hours = 8.3km

She now only has 1.7km to get to her destination.

However, there has been a crash on the motorway so everyone is looking at the cops and the crash as they drive past. So her speedometer fluctuates and changes constantly as she accelerates, brakes, accelerates, brakes. Her Velocity is not constant. It takes her 5 minutes to get past the crash. Sometimes her speedometer, her instantaneous velocity is at 5km/h, then its back down to 0km/h. But we can find out her average velocity.

v= d/t

v = 1.7km/0.083h

v = 20.5 Km/h

Kilometers per hour ? Or, meters per second?

Kilometers per hour is the unit that we are most familiar with, as it is in our cars.

The reason we use it in cars, is it makes it easier to calculate how long it will take you to get to your destination

However, in Physics we use meters per second

To move between the two, remember that there are 1000 meters in a Kilometer, and 3600 seconds in an hour

So 1 km/h is the same as 1000m/3600s = 0.28 meter per second. And 100km/h is 28 m/s

Likewise, 1m/s is the same as (1m x 60s x 60 min) /1000 m = 3.6 Km/h

To make it simpler, to convert meters per second to Km/h, just times the value by 3.6

If I run at 8 meters per second, then that is the same Instantaneous Velocity as 8*3.6 = 28.8 km/h

Likewise, if you have a Instantaneous Velocity of 28.8km/h and you want to find out what it is in meters per second, just divide by 3.6.... 28.8 km per h / 3.6 = 8 meters per second

Side note:

ms^-1 means m/s

and

kmh^-1 means km/h

The reason it is written that way is that it gets the divided by symbol out of the way otherwise the equations can get a bit confusing

Mr Cowley Lectures: Velocity

Mr Cowley Lecture 1 - Velocity

Mr Cowley Lecture 3 - Is it Speed or Velocity?

Mr Cowley Lecture 2 - Triangles and Conversions

Mr Cowley Lecture 4 - Types of Velocity

SciPad

Activities for Velocity: Pages 10, 11 & 12

Answers

Acceleration

Acceleration = Δ Velocity / Δtime

To calculate Δv you need to take the final velocity and minus the initial velocity

Δv = velocity final - velocity initial

Δv = vf - vi

The same applies to finding out the change in time

Δt = t final - t initial

To accelerate is to change your velocity

This happens when you are are driving in the country and are stuck behind a tractor

You are traveling at 50km/h behind the tractor

Then when the road is clear, you overtake, speeding up to 60km/h in 2 seconds

What was your rate of acceleration?

Well when you were stuck behind the tractor, your acceleration was 0km/h/s. However when you overtook the tractor your rate of acceleration was 5km/h/s

How is this so?

Well we use the change in Velocity and then divide it by how long it took you to change that velocity

We can simplify this to:

Acceleration = change in Velocity / change in time

The illuminati stands for change. Their symbol is the greek letter Delta

Here, delta means change

So we can use this letter to further simplify the equation

Acceleration = Δ Velocity / Δ Time

If you are on the motorway at 100km/h and 10 seconds later you are still at 100km/h then your acceleration is 0km/h per second. There has been no acceleration or deacceleration

When you are sitting in the car, you only feel pushed back into your seat or thrown forwards when the motion changes - only when the car accelerates or deaccelerates

Imagine being in a dragster as it accelerates!!

Have a look at the second clip "Xtreme 3 minute Videos"

Here we have a Kiwi racing in Sydney

We can work out the rate of acceleration for the 'worlds fastest Toyota 2JZ dragster'.

The run we will use is starts at 2 minutes 42 seconds on the clip

After the run the big digital readout will tell you the information

However, it is also in the video description

Initial Velocity = 0 kilometers per hour (because its not moving)

Final Velocity = 413km/h

Time = 5.7 seconds

If we put this all into our formula we will get

a = Δv / Δt

Acceleration = (Final Velocity - Initial Velocity) / time

Acceleration = (413 km per h - 0 km per h) / 5.7 seconds

Acceleration = 413 km per h / 5.7s

Acceleration = 72.5 km per hour per second

Acceleration = 72.5 km/h/s

This means that every second that passes the velocity of the dragster increases by 72.5 km/h.

Acceleration: 72.5km/h/s. With this acceleration value, we can plot the Velocity at each point in time as seen below

Time in seconds: 0s 1s 2s 3s 4s 5s 5.7s

Velocity reading on speedometer: 0km/h 72.5km/h 145km/h 217.5km/h 290km/h 362.5km/h 413km/h

The calculation for Acceleration will give you the rate at which an object changes speed.

It will tell you how much the Velocity changes in each unit of time

It will tell you how much your velocity changes in each Second

Mr Cowley Lectures: Acceleration

Mr Cowley Lecture - Acceleration Part 1

Mr Cowley Lecture - Acceleration Part 2

I had 2 cars. A 2016 V6 Commodore, and a 1998 Nissan Pulsar.

They can both hum along nicely at 100km per hour

The big difference is going from 0km per hour to 100km

With the accelerator pushed to the floor

My Pulsar, could do it in 10 seconds

My Commodore can do it in 5 seconds

So, the Commodore will reach the 100km/h mark sooner than the Pulsar

If I was watching and timing the speedometer in the Pulsar, then each second that passes, the speedometer would read:

= Pulsar, km/h: 0, 10, 20,30,40,50,60,70,80,90,100 km/h

Whereas in the Commodore, the speedometer per second would read:

= Commodore, km/h: 0, 20, 40, 60, 80, 100 km/h

We can quantify this rate of acceleration by looking for the pattern in the change in the speedometer for each second.

We can see that the Pulsar's speedometer increases by 10 km/h per second and the Commodore's speedometer increases by 20 km/h per second.

This means that the Commodore has a higher rate of acceleration than the Pulsar

But can this be proven mathematically?

acceleration = change in velocity divided by change in time

acceleration = Δvelocity /Δ time

a = Δv/Δt

Pulsar = 100 kmh /10s = 10kmh/s

Commodore = (100 kmh) / 5s = 20kmh/s

So with the accelerator pushed down to the floor, the Commodore has a higher rate of acceleration than the pulsar

So, why is the accelerator pedal called the accelerator pedal???

This clip shows the relationship between Acceleration, Velocity and Distance travelled

Acceleration = Red

Velocity = Blue

Green = Distance

Notice that Acceleration is 0 whilst the velocity is constant

Velocity only changes if Acceleration is not 0

The bigger the acceleration value the steeper the change in velocity

Also notice that acceleration can be negative!

Negative acceleration is deacceleration

So when acceleration is negative, the car is slowing

In the clip above, a couple of my former students work out their rate of acceleration for their go-kart ride

In the clip below, a couple of my former students work out the rate of acceleration for their bike ride

SciPad

Activities for Acceleration: Pages 18, 19

Answers

Distance-Time and Velocity-Time Graphs

Distance time Graph - Specific Learning Objectives

Draw distance/time graphs from data obtained in motion experiments, including those using ticker timers
Describe, from the shape of its distance-time graph, the motion of objects that are stationary, travelling at constant speed, accelerating or decelerating.
Find the speed of an object from the gradient at a given point on a distance-time graph.
Calculate the distance covered by an object using a speed/time graph of its motion.

The phone - watching you

Google and your phone track your movements

(go to google maps and click on timeline - it shows you where you have been!!).

Take this morning

You live next door to the school, the gate is only 100m from your front door:

You walked a distance of 50 meters in 50 seconds

This is v = d/t = 50m/50s = v = 1m/s

Then you stopped to tie up your shoe lace, so the line at '2' is flat. This means you are stationary and that your velocity is = 0 m/s
Then you continued your walk to school, but at a slower pace, this is can be seen as the line is less steep.

The velocity can be calculated by taking any 2 points along that line, say from seconds 110 to 150 (a change of 40 seconds) and then measuring the distance traveled during that time (80m - 60m = 20 meters)
Then, v = d/t
v = 20 m / 40s
v = 0.5 m/s

You arrive at school, 100m from your house, you rest again for a few seconds, as can be seen by the line being flat again.

While there, you realize that it is Saturday, and the school is closed

You start the walk home, very slowly at first, but speeding up as you realize that you can go back to bed

the graph is sloped here, i.e. it is curved, showing an increasing rate of displacement - in other words your speed is increasing, that is you are accelerating

The graph is a straight line again, indicating constant velocity. This is also the steepest section of the graph, showing that you are at you are covering the most meters per second here than at any other part of your journey.

The change in distance during the 10 seconds between time 260 and time 270 is 20 meters. So, the velocity is:
- v = d/t
- v = 20m / 10s
- v = 2m/s

You are home again, walking towards the door, you reach into your pocket to get your keys, slowing your stride as you approach the door to insert your keys. Here the line breaks and becomes a curve again, the curve flattens over time. This shows that you are slowing down or decelerating
The distance is 0, but time continues. This shows that you are no longer moving:

The change in distance is 0m - 0m = 0m
The change in time is 360s - 340s = 20s
Velocity = distance / time
Velocity = 0 m / 20s
Velocity = 0 m/s

This exact same story can be told as a Velocity - Time graph as shown below

Because velocity is a vector, the positive values mean the velocity away from your house and towards the school, whilst the negative values mean the rate of return to your house

Mr Cowley Lectures: Graphs

Distance Time Graphs Part 1

Distance Time Graph Part 4

Distance Time Graphs Part 2

Velocity Time Graph

Distance Time Graph Part 3

All Graphs with Real Data

SciPad

Activities for Distance-Time and Velocity-Time Graphs: Pages 13, 14, 15, 16, 17, 20, 21, 22, 23, 26, 27

Answers

Gravity and Galileo's Ramp

Specific Learning Objectives

Draw speed/time graphs from given data or data obtained in motion experiments.
Solve problems using the acceleration due to gravity where g = 10ms^-2 near the Earth’s surface

Gravity

Gravity.... It is what holds us on our planet

So effective is gravity, that we don't feel our earth as it spins around at 1600km/h whilst orbiting the sun at 108000km/h

Gravity is the attraction between two objects. Between the earth and ourselves.

All objects generate gravity

The mass of the object is very important. Gravity is a weak force, if you put two bowling balls next to each other, their gravitational attraction to each other is nothing noticeable.

it is only noticeable when the object is big - stand next to a 300m container ship and you will feel....nothing.... the container ship is not big enough.

Stand next to the Burj Khalifa and you will feel.... well, still nothing, still not big enough

Stand on the moon... ok now you are felling gravity. You can jump, you can jump really high on the moon, but it has enough gravitational force to pull you back down.

Stand on earth, that is much bigger than the moon, so it has a gravitational force that is about 6 times that of the moon.

Stand on Jupiter, which is much, much bigger than the earth, and you may be able to stand briefly - but you body will be 2.5 times heavier than it is on earth -much like trying to piggy back The Rock

Stand on the sun, which is massive, (you are the flame from the fantastic four and heat doesn't effect you) and you'd be instantly crushed as you would be 28 times heavier - so me, at 80kg, would suddenly become 2240kg - More than placing my Holden Commodore on my shoulders.

How big do you need to be for gravity to be noticeable

At 1476 Trillion kg, and just under twice the size of Mount Everest, the Mars moon of Deimos is large enough to generate enough gravity for you to walk on it, however, do not jump! Deimos does not have enough gravitational force to return you to the surface if you do a running, so you can run, jump, and drift off into outer space

At 10658 Trillion kg, and 11 times the size of Mount Everest, the Mars moon of Phobos is large enough for you to not be able to jump off it. However, if you drive your lunar rover at over 50km/h up a ramp, then you can self launch into outer space!

So...size matters! The bigger the object the more gravity. All objects generate gravity, you generate gravity, but it takes extremely large objects for the gravity to be noticeable. Also, you need to be close to the object, the gravitational pull of a planet increases the closer you get to it.

If you jump out of a plane, you will accelerate towards the earth. This is important. The further the object gets to fall, the faster it is going. If you drop a ball from head height, and your friend drops a ball from the top of the your house the ball from the roof will be traveling faster just before it hits the ground. Likewise, if you jump of your chair, and your friend jumps from your roof, your friend will be moving faster, they will have a faster velocity, when they hit the ground then you will be. Thus, they will break their legs.

How Fast is Acceleration due to Gravitational Force on Earth

How fast do objects accelerate towards the ground? This was a problem that Galileo thought about, he decided to calculate the distance traveled in each second by the object as it fell, then convert this into velocity per second, and then calculate the rate of change in velocity per second. However, although he dropped objects off the Leaning Tower of Pisa, the objects were moving too fast to be able to measure their velocity.

Like falling, if something goes down a ramp, it will go faster and faster. So, instead of measuring free fall, he decided to measure the rate at which an object accelerates down a ramp and then use trigonometry to calculate what this acceleration would be if the object had been in free-fall.

An object will accelerate at a rate of 9.81m/s per second, towards the earth, ignoring air resistance. So, at the end of the first second, the object will be traveling at 9.81 m/s, at the end of the second second, the object will be traveling at 19.62 m/s, at the end of the third second, the object will be traveling at 29.43 m/s.

In theory, all objects should fall at the same rate, however air resistance slows objects down - which is why feathers fall slowly in the air - but they will fall at the same rate as a rock in a vacuum. It is because of this air resistance, that car designers will try to make their cars aerodynamic, to reduce drag

Graphing Gravity - Constant Acceleration

Gravity Graphing

Acceleration due to gravity is great for graphing because it is a constant rate of acceleration, which leads to a linear increase in velocity and a exponential increase in total distance traveled.

Galileo's Ramp

Galileo was trying to show that objects accelerate as they fall to the earth, and to calculate that rate of fall.

However, as you would know when dropping a ball or an object from a height, it moves very quickly.

This was a problem for Galileo, so he figured that if he could slow down the ball a bit, then he could more accurately observe the rate of acceleration.

To slow the ball down, he let it fall down a ramp. As the ball rolls down the ramp, it will accelerate at a constant rate, which is what happens when the ball is dropped vertically. Just the rate of acceleration isn't as extreme. This allowed for more precise observations.

You can easily make your own Galileo's Ramp and do his experiment - all you need is a length of timber, some rulers, and a slow-mo camera (he didn't have one of these in 1638)

When you look at the graphs created using Galileo's ramp, you'll notice that they have the same pattern as the graphs used creating actual falling objects

Thanks to Kyle, Ronak and Anne. MHJC, class of 2018

SciPad

Pages 20, 21, 22, 23

Answers

Gravity Part 1

Gravity Part 2

Mr Cowley Lectures about Motion

The lectures below are the same content as would have been delivered on a whiteboard in a classroom.

They refer to this webpage, there are differences in the layout as I have updated the webpage after making the videos