Speed, Velocity and their Averages

Transcript

0:00Introduction

Hello everyone! First, one small puzzle before we start. Here is a car. It goes sixty kilometres from P to Q at thirty kilometres per hour, and it comes back at sixty kilometres per hour. Now tell me, what is its average speed for the round trip? Thirty and sixty, so forty five, right? No, no, wait. Many students say forty five, and it is wrong! By the end of this video, you will see exactly why. And look, speed and velocity are the first two words of kinematics, and almost every kinematics problem uses them. So, let's begin.

0:40Signs on a straight line

So, motion along a straight line. Before anything else, we choose an origin, our zero point, and a positive direction, okay? Say, towards the right is positive. Clear? Now the position of the particle is x, and the change in position, the displacement, is delta x. That small triangle is the Greek letter delta; it means change in, okay? Both are signed numbers: the sign tells you the direction. Alright, so if a particle's velocity is negative, what does that tell you? Yes, simply that it is moving in the negative direction, nothing more! Velocity v and acceleration a are signed numbers too, okay? On a straight line, the sign does the job of the direction. That's it!

1:27Average speed and average velocity

Now the two averages. Average speed equals total distance divided by total time. Distance means the full path length, every metre you travel, forward or back. Now, average velocity has the same total time at the bottom. But what goes on top? Yes, total displacement! Average velocity equals total displacement divided by total time. Displacement only cares about where you started and where you ended, and it carries a sign. Alright, now think. Which one is bigger, average speed or average velocity? Average speed is always greater than or equal to the size of the average velocity; that is the sign on screen. The two straight bars mean mod, the size without the sign, okay? Why? Because the distance travelled can never be less than the size of the displacement, right? And the two become equal when the particle never turns back. Look, in a full round trip, the displacement is zero, so the average velocity is zero, but the average speed is not zero. That is a classic trap!

2:33Equal distances: the harmonic mean

Next comes the most important case. Here, a body covers two equal distances, each d, one at speed v one, and the other at speed v two. The small one and two at the bottom are just labels, okay? Now, can we simply add the two speeds and divide by two? No, no! Go back to the definition: total distance over total time. Time for each part is distance by speed, so total time t equals d by v one, plus d by v two. Got it? The total distance is two d. So average speed is two d by t, and the d cancels to give two v one v two, over v one plus v two. Write this down! This is the harmonic mean of the two speeds, not the plain average, v one plus v two by two. Why? Because the body spends more time at the slower speed, right? So the slow part pulls the average down. For three equal distances, the same idea gives three v one v two v three, over v one v two, plus v two v three, plus v three v one. Look at the bottom: each pair of speeds multiplied, going round in a circle. With me?

3:45Equal times: the arithmetic mean

Now let's flip it. This time the body moves for equal times, t each, first at v one and then at v two. Which average do you get now? Don't guess, work it out! Distance is speed into time, so the distance is v one t, plus v two t, covered in a total time of two t, okay? Divide, and the t cancels. Average speed equals v one plus v two, by two. Yes, this time the plain average, the arithmetic mean, is correct! So here is the topper's way to remember it. Equal distances: harmonic. Equal times: arithmetic. In the question, look for that one word, distance or time. It decides the formula. Clear?

4:31Topper's trick: changing the units

One more tool, and toppers use it all the time. The k m h to the power minus one is just kilometres per hour, and m s to the power minus one is metres per second, okay? One kilometre per hour is one thousand metres in three thousand six hundred seconds, that is, five by eighteen metres per second. So, quickly tell me. Seventy two kilometres per hour is how many metres per second? Seventy two into five by eighteen: twenty metres per second! Learn these four by heart: eighteen, thirty six, fifty four and seventy two kilometres per hour become five, ten, fifteen and twenty metres per second. Believe me, they save time again and again.

5:16Example 1: round trip at 30 and 60 km/h, not 45

So, remember our puzzle? Let's settle it now. Here, a car goes sixty kilometres from P to Q at thirty kilometres per hour, and returns at sixty kilometres per hour. Find its average speed and average velocity for the round trip. Pause the video, and try it first. Done? Okay, let's check. Going takes sixty by thirty, two hours. Coming back takes sixty by sixty, one hour. Total time, three hours; the h on screen means hours, okay? Total distance: sixty there, plus sixty back, one hundred and twenty kilometres. So, average speed is total distance by total time. What do you get? One hundred and twenty by three: forty kilometres per hour! Not forty five. See, mystery solved: the car spends two hours at the slow speed and only one hour at the fast speed, so the average comes down to forty. And check it with our formula for equal distances: two into thirty into sixty, by thirty plus sixty. Three thousand six hundred by ninety: forty again. Exactly the same! Now the second part. What is the average velocity for the round trip? Yes, zero! The car comes back to P, so the displacement is zero, and so is the average velocity. Same trip, two very different answers. Interesting, right?

6:43Example 2: three equal parts at 20, 30 and 60 km/h

Now a slightly harder one. Here, a car covers three equal parts of a journey, at twenty, thirty and sixty kilometres per hour. Find its average speed. Pause and try. Hint: equal distances, so think about time. Here is what many students do: twenty plus thirty plus sixty, and divide by three. Correct, right? No, no, careful! The parts are equal distances, not equal times. So call each part d. The time is d by twenty, plus d by thirty, plus d by sixty, and the total distance is three d. Got it? Average speed is three d by the total time, and the d cancels: three, over one by twenty, plus one by thirty, plus one by sixty. Now add these three fractions, with sixty as the common denominator. What do you get? Three by sixty, plus two by sixty, plus one by sixty: six by sixty. And look at the three, two and one on top: the car spends three times as long at twenty as at sixty. So average speed is three, divided by six by sixty: thirty kilometres per hour. Less than the plain average, because the slow part takes the most time, right? Same idea as our puzzle.

8:03Quick revision: harmonic mean, arithmetic mean, 5/18

So, quick revision. Average speed is distance over time; average velocity is displacement over time. Equal distances, harmonic mean. Equal times, arithmetic mean. And from kilometres per hour to metres per second, multiply by five by eighteen. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!