Physics
Velocity and Acceleration
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
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. :-)
Instantaneous Velocity vs Constant Velocity vs Average Velocity
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
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
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
Also why is the accelerator pedal called the accelerator pedal???
This clip above 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 the rate of acceleration for their bike ride
In the clip above, a couple of my former students work out their rate of acceleration for their go-kart ride
