Speed

Miguel Aveiro
Jul 11, 2020
9 min read

Updated: Feb 28

This article covers:

Why talk about speed?
How to calculate speed
Velocity
Scalar quantities and vector quantities
Instantaneous and average speed and velocity
Acceleration
Momentum

Why talk about speed?

I created this article on speed for three reasons: 1) to understand how to calculate the speed of objects or the duration or time of a journey; 2) it's in school physics, so possibly, I might explain it in a way that gives you a better understanding than you would get in school; 3) it helps you understand other subjects in physics, including the following article on forces.

How to calculate speed?

Speed is moving a certain distance in a certain amount of time. A higher speed will not change the distance, but it will make the journey take less time. Also, the longer the distance, the faster you need to go in order to travel that distance within a set time. For example, you have 5 minutes to get to your destination. One route is 100 metres in distance, another is 300 metres. You will need to travel faster to cover 300 metres within 5 minutes than 100 metres in 5 minutes. Basically, you need to move 3 times faster if your route is 300 metres!

The formula to calculate speed is:

speed = distance/time (distance divided by time)

So you're dividing the distance by the time it takes.

We'll use our example of traveling 300 metres in 5 minutes. First, we need to sort out the units. The metres for distance can stay, but we'll change the minutes into seconds: 5 minutes = 300 seconds. That way we'll have metres per second, or m/s as our unit for time. We could say how fast we're going in terms of minutes, but scientists need a standard unit instead of changing it all the time from seconds to minutes, to hours etc. Metres per second is the standard unit.

Okay, so 300 metres in 300 seconds = 300/300 = 1 m/s. Every metre takes only 1 second to cover. If instead of 300 metres, we traveled 600 metres, this will be: 600/300 = 2 m/s.

You can think of distance as the 'big thing' that's being divided. If you have a cake and it's being divided by 6 individual people (or 6 people each get a slice of cake) the cake is the 'big thing'.

So:

slice of cake = whole cake/individuals

If the cake is made bigger, then each person will get a bigger slice. If the cake is smaller, then each person will get a smaller slice.

Also, if you had more individuals but kept the size of the cake the same, each slice will be smaller. Less individuals will mean bigger slices.

Going back to speed: distance is the 'big thing' being divided by amounts of time, to get how much distance each amount of time takes. So how many metres out of the total distance does each second of time get? So if you increase the distance, the more metres each second gets (more metres per second) and conversely for decreasing the distance (less distance, less metres per second). If you increase the number of seconds, the less metres each second will get and conversely again for decreasing the seconds.

The reason for this long explanation, is so you can understand why you divide distance by time in order get the speed, instead of dividing time by distance or multiplying both together. I want you to get a complete understanding. This will help when doing any other formula in the future.

If you want more help as to why you don't divide the time by the distance, then imagine instead of saying you're traveling 50 km/hour (kilometres = 1000 metres) you'd say 50 hours/km. You'd be saying each km takes 50 hours. Then try working out how fast you're going if you increase the km or the hours. Anyway, you don't have to try that. We can move on.

Instead of typing out speed, distance and time, we can abbreviate them as 's', 'd' and 't' respectively.

Therefore:

s = speed (in metres/second)

d = distance (in metres)

t = time (in seconds)

Rearranging the formula

Suppose instead of speed, you want to work out the distance or the time instead. For this, we need to assume we know the speed. So, if we're traveling 50 m/s and we want to know how much distance we can cover in 250 seconds, we would need to re-arrange the equation.

First, let's look again at the equation (formula) to get speed:

speed = distance/time

We think of the '=' as separating the two sides of the equation. You work out one side to get the other side. If we're working out the speed, we need it to be on its own on one side of the equation. So, we work out the right hand side, in this case by dividing distance by time, and then we obtain the answer, in this case the speed.

We want distance instead, so we need to do something to time and speed. The trick is to know that if you "do something" to one side of the equation, the same needs to happen on the other side. By "do something", I mean multiply, divide, add or subtract. We don't need to add or subtract in this equation, but I'm just saying giving your options for equations in general.

Okay, our current formula has distance being divided by time. We need to reverse that to get distance on its own. So we multiply the right side of the equation by time and we do the same to the other side:

(x time) speed = distance/time (x time)

speed x time = distance

(x time) is the opposite of (/ time) so we can see that they cancel each other out on the right hand side of the equation.

So, let's punch the numbers in to get our final result:

50 m/s x 250 s = 12,500 m.

So, if we're traveling at 50 m/s we can cover 12,500 metres in 250 seconds.

We did that in one step. We just multiplied both sides of the equation by time.

If we want to get the time instead, we can't do it in one step. If what you want is past the dividing line, you first need to get rid of the dividing line. So we follow the same step as to get the distance, which is to multiply both sides of the equation by time. So, we're currently here:

speed x time = distance

Now, to get time on its own, we need to get speed over to the other side with distance. We need to reverse the multiplying by dividing. So:

(/ speed) speed x time = distance (/ speed)

---->

time = distance/speed

Again, we see two opposites cancel each other out: (x speed) and (/ speed).

You can rearrange any formula like this to find the different values.

Velocity

Velocity may sometimes be confused with speed. You may hear that velocity is speed in a certain direction but a better way of putting it is the rate at which an object changes its position. If you move forward 1 metre in 1 second and then move back 1 metre in 1 second, your speed is 1 m/s but your velocity is zero.

In that 2 seconds of time, you have not been displaced from your original position. If you only counted the first second when you moved forward, you would have velocity as you were displaced.

So how much velocity? For that we need to add in the direction. It's not enough to say you moved 1 metre, but 1 metre forward. But what is forward? If you were on the phone with someone who is walking in a field and they said they walked forward, how do you know the direction they actually walked? Forward is relative to which direction they're facing. If they're facing north, then they walked north. If you were giving them directions, you would say where they would go in compass directions. Their velocity will include how fast they're going and it will depend on what you're counting as your starting position.

Just as speed is abbreviated as 's', velocity is 'v'.

Scalar and vector quantities

So let's back up here. When we were thinking of speed, this was a quantity of magnitude. The number was simply an amount of something, in this case an amount of speed. You can scale it up, thus increasing the number; or scale it down, thus decreasing the number. Therefore, we call this type of quantity scalar.

With velocity, we can't simply scale the quantity up or down. Sure, you can with part of it; that is, you can scale the speed. However, you can't have more north or more west. North and west are vectors, or directions. We call velocity a vector quantity. Such a quantity can be positive (+) or negative (-). For example, if you moved 1 metre north at 1m/s and then 2 metres south at 1 m/s, then you've moved -1 m/s north.

Any other principle that uses velocity, such as acceleration and momentum, which are detailed below, are also vector quantities.

Instantaneous and average

If you move 1 metre forward (or north) during the first second and then 1 metre back (or south) during the following second, then the average velocity during the whole two second journey is zero. However, if we take just the first second, your instantaneous velocity, or your velocity at that instant, was 1 m/s north.

The average velocity is the sum of all the velocities for the entire journey. Two velocities with opposite directions cancel each other out; so our example left us with an average velocity of zero.

We can do the same for speed (and it would be easier without directions!) If you were traveling at 6 m/s during the first second, 2 m/s during the next second and 4 m/s during the last second, this will give us three different instantaneous speeds. We can then work out the average speed by adding them together and dividing by 3, leaving us with 4 m/s.

Acceleration

When we talk about the speed of an object, we actually mean its average speed over a length of time and over a certain distance. In reality, however, an object will be speeding up and slowing down over time. As well as speed, we can calculate the acceleration of an object. However, acceleration, as the concept is used by physicists, is actually a change in velocity over time.

So we take the object's velocity in m/s (+ direction) and then divide it by the amount of time.

If the object is accelerating, then at the end of the journey it will be faster than it was at the beginning. The reverse is true if it was decelerating. So we work out how much the velocity has changed. For simplicity's sake, we'll keep the direction the same throughout the journey and so ignore it for now. If the object was going 2 m/s at the start and then 5 m/s at the end, then the speed has changed by 3 m/s.

We need to know how many seconds that took, because if it took 3 seconds to make an increase in 3 m/s, then that's a greater acceleration than if it took 9 seconds. So if it took 3 seconds, then the acceleration was 3 m/s in (or divided by) 3 seconds = 1 m/s/s. Each second, the object accelerated by 1 m/s and so in 3 seconds it had accelerated by 3 m/s.

Now, if it took 9 seconds to accelerate from 2 m/s to 5 m/s, then the acceleration would be: 3 m/s divided by 9 seconds = 0.33 m/s/s.

Instead of writing m/s/s, you can simplify it to m/s2 (sorry, I can't do superscript on this website, but you can see it written correctly in the images above). Don't forget to add the direction at the end, to make it a change in velocity.

We can have a change in direction, with different speeds in each direction, like we've seen before, and therefore a change in acceleration in different directions. Come up with examples of this yourself.

Momentum

The term momentum is used in association with movement. You may have heard that a vehicle is gaining momentum or figuratively when talking about a popular idea gaining momentum.

When we discuss momentum, we're not just talking about speed but it depends also on the object itself. A more massive object has a greater momentum than a less massive one. So when a large truck is moving along, it has more momentum than a small car.

Also, again, momentum refers to velocity rather than speed. So if an object moves forward 1 metre at 1 m/s and then back 1 metre at 1 m/s, then the overall momentum is zero.

Momentum is a vector quantity and it also has instantaneous and average values, just like with speed, velocity and acceleration.

The equation for momentum is:

momentum = mass x velocity

It can also be written as: p = mv

where 'p' is momentum, 'm' is mass and 'v' is velocity. You don't need to put 'x' for 'times' or 'multiplied by' as when you put two letters together, it means they are multiplied together.

As for the units, mass, for more massive objects, is in 'kg' and velocity is in 'm/s in a direction'. Then you combine the two for momentum: kg m/s (+ direction).

For example, if you have a 40 kg object moving north at 3 m/s, then the momentum is:

p = mv

p = 40 kg x 3 m/s north

p = 120 kg m/s north

The momentum of an object can change if: