After how long, Ram hears a splash?

when Rahm was dancing on a bridge his phone suddenly drops off and into the river 180 metres below after how long will he hear a splash given the speed of sound is 340 meters per second and the acceleration due to gravity G is 10 meters per second squared so let's draw the situation first so here's my good friend Rahm who usually likes to dance on a bridge and suddenly his phone falls off it's given it falls into the river 180 meters below therefore this phone will fall down 180 meters below so clearly this picture is not to scale and the question is after how long will he hear a splash so when the phone falls into the river like this it'll make a splash sound and we need to figure out after how long will drum hear the sound of that splash so how do we do this well my first thought is something like this since we want to find out how long it takes to hear the sound basically we just have to calculate how long it takes for this phone to go into the river right I mean let's say the phone takes five seconds to fall into the river this is an example then it'll take five seconds for RAM to hear that sound right No and that's because sound takes time to go from one point to another so when the phone hits the river a sound is created but that sound does not reach RAM instantly it has to travel all the way back up and so that's going to take some additional time so let's say that sound takes about I don't know as an example we'll have to calculate but let's say it takes about 0.2 seconds to travel back up then Rahm will hear the splash after 5.2 seconds five seconds for the phone to hit the water and an additional point two seconds for that sound to travel upwards and so that's pretty much what's going on over here so this means we need to do two calculations first we need to figure out exactly how long it takes for the phone to hit the river the five second was just an example and then we need to calculate how long does the sound take to travel all the way up and the total time will be the sum of those two so now that we have a rough plan of what we have to do great idea to first pause the video and see if you can try solving this yourself first okay let's do this so let's first calculate how long it takes for his phone to hit the river so let's see what's given to us it's given that his phone suddenly drops off so that phone drops off from Rams pocket and whenever it's mentioned something is dropping it means its initial velocity is zero so we can say the initial velocity of this phone which we usually write as U is zero what else do we know we know it's falling into a river a hundred and eighty meters below this means we know the displacement of the phone before hitting the river it would have displaced a hundred and eighty meters so s which is used for displacement is a hundred eighty meters okay what else well as this phone is falling down its speed keeps increasing right and so we know it is accelerating and since it's a falling under gravity we know its acceleration its acceleration due to gravity that's given to us so the acceleration of that phone must be equal to G which is ten meters per second square usually we take it as 9.8 sometimes in problems they ask us to assume it to be ten but one question we might have over here is whether this is positive or negative because accelerations can be negative as well how do we decide that well in general whenever any object is increasing its speed that's when we say its acceleration is positive it's gaining speed on the other hand if the object is decreasing its speed losing speed that's when we say it has negative acceleration or deceleration so let's see what's happening over here well our phone is falling down and we know that our phone is speeding up right and as a result this acceleration is positive because it's gaining speed but for instance if Ram had thrown his phone up for some reason then as it goes up its speed would decrease right it would decrease it will slow down and they'll fall back that time during the upward motion we would say acceleration is negative so for our downward motion acceleration is positive and we need to calculate time this is what we want to calculate so we need an equation that connects these four quantities and you might have learned that there are three equations of motion we equals u plus eighty s equals UT plus half a T squared and V squared equals u square plus 2ei us which one do we use well let's look at them one by one if you look at the first one we have V which is the final velocity over here that will be the velocity with which our phone hits the river that's something we don't know you we know a we know T also we don't know so in the first equation there are two unknowns two things which we don't know and we can't do any calculations so the first equation is useless let's get rid of that let's look at the second one we have s which we know U which we know T is unknown a we know T is the same unknown hey this equation only has one unknown which means we can go ahead and use this equation and we can get rid of the last equation as well because it does not have T at all in it so there's no point in looking at it so we have a winner we can go ahead and use the second equation and you know what if you didn't solve this before again now would be a great time to pause the video and see if you can try this yourself see if you can substitute the values and find the value of T from this all right let's do this s is 180 I won't write the unit's because all units are standard and so the the unit of T will also be standard second so I won't try it okay s is 180 U is 0 so this whole term will be 0 so I won't write that so 0 plus half a which is 1010 T squared T squared and now we just have to do the algebra to get T squared so let's see 2 goes 5 times so this goes 5 times over here and now to calculate T Square I want to get rid of this 5 from the right side so I'll divide by 5 on both sides so let me divide by 5 here and divide by 5 over here 180 divided by 5 is 36 you can check that 5 goes 3 times 15 and if I go 6 times 30 so 36 equals T Square this gets divided out over here T Square and this means T is the square root of 36 so T is the square of 36 which is 6 second and there is our answer this means the phone takes 6 seconds to go and hit the so you've done the first part now we need to calculate how long it takes for that sound to travel upwards so let's do that over here how do we do that do we use the equations of motion again well we don't have to and that's because sound is unaffected by gravity so sound will not accelerate or decelerate it will travel with a constant speed of 340 meters per second as it travels upwards this actually makes things simpler for sound since it's travelling with a constant speed we can always use the formula speed equals distance by time this works for anything that travels with constant speed so for sound all we'll do is speed so speed equals distance divided by time we're doing this because it has no acceleration it's traveling with a constant speed so again we just have to plug in we know the speed of sound that is 340 so let's plug that in this is 340 that's equal to distance the sound has to travel the same distance 180 divided by T and so now if you do the algebra teve turns out to be a hundred and eighty divided by 34 340 and if you do the long division which I leave to you so that we can save some time over here but if you do that long division the answer turns out to be roughly 0.53 and so that is how much time it takes for sound to travel upwards sound is pretty fast so it travels pretty quickly only about 0.53 seconds and so this means that the total time it takes for arm to hear the splash would be the sum of these two so the total time let's use a different color for that let's use this color so the total time that's going to be six plus point five three that's going to be six point five three seconds and so this means that ROM will hear the splash six point five three seconds after the phone falls off although knowing ROM I'm guessing it's gonna take him a lot longer to realize he has lost his phone