Relative Velocity in One Dimension

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Two trains on parallel tracks, one fast, one slower. When the fast one overtakes the slow one, it takes a long, long time. But when the same two trains cross from opposite directions, whoosh, it is over in seconds. Same trains, same speeds. So why such a big difference? Today, relative velocity answers this. And remember, in JEE, chase, overtaking and lift problems are all built on this one idea. So, let's begin.

0:35Velocity depends on who is watching

So, picture two cars, A and B, on a straight road. You sit in car A, and watch car B, okay? Does car B seem to move with its own full velocity, or with something else? Yes, something else: only the difference of the two velocities! We write it like this: v B A vector equals v B vector, minus v A vector. The little arrow on top means a vector. And the small B slash A below v means, B with respect to A, okay? So, the velocity of B, minus the velocity of the watcher. Clear? The same rule works for position and acceleration. r B A vector equals r B vector, minus r A vector; r is the position vector. And a B A vector equals a B vector, minus a A vector; a is the acceleration. Always: the body you watch, minus the body you sit on. What does this really mean? It is how B moves, as seen by an observer riding on car A, without rotating. For that observer, car A is simply standing still. Got it? Alright, now swap seats. You sit in car B, and watch car A, okay? How does car A seem to move now? Yes, just the opposite! v A B vector equals minus v B A vector. Same size, opposite direction, okay? If car B seems to move forward for you, then for car B, you seem to move backward.

2:03On a straight line: signs do the work

Today we stay on one straight line, where a sign does the job of the arrow. Choose one direction as positive; velocities the other way get a minus sign. Then v B A equals v B minus v A, with the signs. Okay? If both bodies move the same way, both velocities have the same sign. So the subtraction gives just the difference of the speeds. See, that is the slow overtaking in our puzzle. Now the two bodies move in opposite directions. The formula has a minus sign in it, so again we take the difference of the speeds, right? No, no, careful! This is the JEE trap. Let A move at speed v one, the positive way, so v A is plus v one; the small one and two are just labels. B moves the other way at speed v two, so v B is minus v two. Then v B A equals minus v two, minus v one, which is minus of the bracket, v one plus v two. The double arrow means, so. The minus sign only gives the direction. The size is v one plus v two: the speeds add! That is why trains crossing from opposite directions are past each other in seconds. With me?

3:15Topper's trick: sit on one body

Now a trick that toppers use. Work in the frame of one body. Climb onto body A, and sit there. In the frame of A, body A is at rest, okay? And body B? It moves with the relative velocity, v B A vector, and the relative acceleration, a B A vector. Just like that, two moving bodies have become one! Now think. Two bodies, both falling freely. What is the acceleration of one, as seen from the other? Zero! Exactly. Both have the same acceleration g, downward. So a B A vector equals g vector minus g vector, which is the zero vector: the zero with an arrow on it. And with zero relative acceleration, the relative velocity does not change. So each one sees the other move in a straight line, at constant velocity. No curve, no speeding up. Very useful in problems, right?

4:10Example 1: trains overtaking and crossing each other

So, remember our two trains? Let's finish that puzzle. Trains of lengths one hundred and one hundred and fifty metres, the m means metres, run on parallel tracks. Their speeds are seventy two and fifty four, in km h to the power minus one, kilometres per hour. How long does the faster take to overtake the slower? And how long to cross, when moving in opposite directions? Pause the video, and try it first. Done? Okay, let's check. First, units. To change kilometres per hour into metres per second, divide by three point six. Seventy two by three point six is twenty, and fifty four by three point six is fifteen, in m s to the power minus one, metres per second. Now use our trick: sit on the slow train, so it is at rest. How far must the fast train move, relative to it, to overtake completely? Only its own length? Both lengths! Overtaking starts when the front of the fast train reaches the back of the slow train. It ends only when the back of the fast train passes the front of the slow one. So, one hundred plus one hundred and fifty: two hundred and fifty metres. Same direction, so the speeds subtract. Twenty minus fifteen: five metres per second. Time is two hundred and fifty by five: fifty seconds; the s means seconds. Opposite directions, so the speeds add. Twenty plus fifteen: thirty five metres per second. Time is two hundred and fifty by thirty five, that is fifty by seven, about seven point one seconds; the wavy equals sign means approximately. There you go, puzzle solved! Fifty seconds against about seven, only because the relative speed changed.

5:59Example 2: ball thrown up inside an accelerating lift

Now a harder one, JEE level. Here, a lift rises with an acceleration of two, in m s to the power minus two, metres per second squared. Inside it, a ball is thrown up at six metres per second, relative to the lift. When does it return to the thrower's hand? Take g as ten. Pause and try. Hint: sit inside the lift. Done? Okay, let's check. Take upward as positive. Once the ball leaves the hand, it falls freely, so its acceleration is g, ten, downward: minus ten. The lift accelerates upward: plus two. The small words below each a just name the body, okay? Now the ball's acceleration, seen from the lift. The lift goes up, so it takes away some gravity: ten minus two, eight, right? No, no, careful! Use the formula, with signs. Here, ball slash lift means ball with respect to lift. So, minus ten, minus bracket plus two: minus twelve metres per second squared. Twelve, downward! The floor rushes up to meet the ball. Clear? Now our trick. In the lift's frame, the thrower's hand is at rest. The ball starts at plus six metres per second, with acceleration minus twelve metres per second squared. It is back in the hand when its displacement s, relative to the lift, is zero. So, zero equals six t, minus half into twelve into t squared. Cancel one t, as t is not zero. Then t equals two into six, by twelve; the dot means into. Exactly one second! With the wrong eight, the answer goes wrong. Signs saved us!

7:47Quick revision: vB minus vA, add or subtract speeds

So, quick revision. v B A vector equals v B vector minus v A vector: the body you watch, minus the one you sit on. On a line, same direction, speeds subtract; opposite directions, speeds add. And sit on one body: it is at rest, and the other has the relative velocity and acceleration. Now try the practice problems on this page. You will find them easy, I promise. All the best!