Main content

### Course: Mechanics (Essentials) - Class 11th > Unit 8

Lesson 3: Can you push a car from inside?- Conservation of momentum
- Momentum conservation derivation
- Conservation of linear momentum (basic)
- Conservation of momentum calculations
- Bouncing fruit collision example
- Momentum: Ice skater throws a ball
- Apply: conservation of momentum calculations

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Conservation of momentum calculations

Conservation of momentum can be used to predict the motion of objects after a collision. Let's try two practice problems together. Created by Mahesh Shenoy.

## Want to join the conversation?

- For the second question, would we have gotten the same answer if we chose to have the other car be moving in a negative direction?(5 votes)
- yes

if two bodies are moving in opposite directions

you can choose any one body to be moving in negative direction.

you will get the same answer.

Here if you chose truck to be moving in negative direction, then--

(2kg)(-4m/s) + (1kg)(17m/s) = 3kgV

-8kgm/s + 17m/s = 3kgV

9kgm/s = 3kgV

V = 3m/s (The magnitude of velocity remain the same, only the sign changes)

In the question he assumed left as negative and right as positive direction..

Here we assumed right as negative and left as positive direction.

Since our answer here is positive, the whole thing will move towards the left. (the same answer as in the video)(5 votes)

- how did he calculate v at the end(2 votes)
- Here's the simplified version. I showed the velocity to the right as positive, and the velocity to the left as negative.

Legend -

M = mass of the truck

U = initial velocity of the truck

m = mass of the car

u = initial velocity of the car

V = total final velocity

MU + mu = (M+m)V

(2 * 4) + {1 * (-17)} = 3 * V

= 8 + (-17) = 3 * V

= -9 = 3 * V

V = -3 m/s

This means that the velocity of the two vehicles together will be to the left; the final velocity will be to the left.

Hope this helps!(7 votes)

- I don't understand how he cancelled the 10 kg from both the sides. Could someone help me out?(2 votes)
- 9kg * 10m/s = 10kg * V

(The 'V' given is a variable)

Now consider this statement as linear equation where the 'V' is the variable. Then what you would do is that transfer the 10kg from the RHS to the LHS leaving the 'V' in the RHS. So when the 10kg comes to the LHS it is going to divide (as signs change when we transfer from one side to the other) so 10 divided by 10 will give 0. So he cancelled the 10s just to reduce the amount of space taken.

Hope this Helps!(6 votes)

- And so whenever we have problems with collisions, we can always start with conservation of momentum.(2 votes)
- In3:20he says that there are no other external forces but there is air resistance. Is that not an external force?(1 vote)
- Yes, air resistance is an external force, but air resistance is friction against air molecules, and the question says to neglect friction. So, air resistance is neglected.(1 vote)

- Is solving non-collision problems the same as the collision problems practiced in this lesson?(1 vote)
- Hello! At10:04, how are you supposed to simplify the equation?(1 vote)

## Video transcript

- [Instructor] Let's solve two problems on collision. Here is the first one. A nine kg monkey jumps
with a horizontal speed of ten meters per second onto a stationary one kg skateboard. With what speed do the monkey and the skateboard move together? Neglect the friction
between the skateboard and the ground. So, let's look at what
is given to us first. Maybe we'll draw a diagram for that. Then, we'll gather the data, we'll see what is asked, and then, maybe, we'll come up with a strategy to solve that. Okay? So, let's first go ahead and
look at what's given to us. We're given that a monkey is jumping onto his skateboard, so here is our picture. So, this is our monkey. He's gonna jump on that skateboard, and then they'll start moving together. What data is given to us? Let's look at that now. We know the weight of that monkey. We know the mass of that
monkey is nine kilograms. We also know the mass of that skateboard is one kilogram, so let's put down these masses. I'm not going to write those individually, that data, individually. To save space, I'm just
writing it on this picture. And we're also given... What else? We are given the monkey jumps with a horizontal speed of ten meters per second. So, this monkey has a speed of ten meters per second, and the skateboard is stationary. So, the skateboard, initially, is at rest. Now, what we need to find out? What do we have to calculate? With what speed do the monkey and the
skateboard move together? So, that means after
jumping on that skateboard, we need to figure out what is the speed with which they move together, so their combined speed is what we need to calculate. So, how do we solve this problem? Where do we even start? Well, I guess, the most
important clue that we can get over here is that this is
a problem on collision, and whenever bodies are colliding, their total momentum before collision should always equal their momentum after the collision, and so that's where we can start. And, just to make that statement a little bit more clear
before we start solving, what's collision? Well, in physics, whenever objects put a force on each
other for a short time, we say they are colliding. For example, when the monkey jumps on this skateboard, it puts a force on that skateboard. It pushes that skateboard forward, which is what accelerates the board, and during the same time, the board pushes back on monkey, Newton's third law. But these forces only
last for a short time. After which, they both start moving with a constant velocity, and therefore, that's a collision and momentum is conserved. And another important thing is that momentum is only conserved provided there are no external forces. If there are other objects besides these two, that
start pushing on them, say, the ground starts pushing on them, then the momentum will not be conserved. It's for that reason it's mentioned in the problem, "Neglect friction," and if you're wondering,
"Well, what about gravity? Isn't that an external force?" Well, yeah, but we don't have to worry about gravity, because it's been balanced. For example, if you consider the gravitational force acting
on this monkey downwards, that's being balanced by an upward push given by the skateboard, and so, the forces cancel out, and we don't have to worry about them. So, if you neglect friction, there are no other external forces, and so, we can see the momentum is conserved. And of course, if you need more clarity on these things, then we have talked a lot about them in previous video, called, "Conservation of Momentum." So, you can always go
back and refer to that. All right, now, I think
we can start the problem, and we will start by saying that the total initial momentum, which
we will write as Pi should equal the total final momentum, momentum after the collision. Now, we know how to
calculate momentum, right? We just multiply mass and the velocity, so all we have to do is calculate the initial momentum of the monkey plus that of the skateboard, and equate it to the final momentum of the monkey and the skateboard, and then, see if we can calculate that velocity. So, you know what? Great idea: You have to
give it a shot yourself. Pause the video and see if you
can first try this yourself. Okay, so, let's go ahead and solve this. So, what's the initial momentum? Well, that's the initial
momentum of the monkey which is the mass of that monkey. I'll use big-sized letters for monkey, and I'll use small-sized letters for the skateboard. I'll also use different colors for them. So, that's when we mass of that monkey into the initial velocity of that monkey plus mass of that skateboard into the initial velocity
of that skateboard. This is the total initial momentum, and that should equal
it's total final momentum. What's that? Again, that's gonna be mass of that monkey into the final velocity of that monkey. Notice the final velocity's the same for both of them, because
they're moving together. Plus, the mass of that skateboard, the mass of that skateboard, into the final velocity
of that skateboard, and before we substitute,
since v is common, because they are going
with the same velocity, we can pull that v out and write is as the total mass, M + m into v, and now we can plug in. We have all the data, and we can plug in. So, let's do that over here. The mass of the monkey is nine kilograms So, that's going to be nine kilograms times u, which
is ten meters per second, plus the initial velocity of our skateboard is zero, because, initially, it was at rest, so this
whole thing goes to zero, so that's our left-hand side. This is our total initial momentum. That should equal - let's look at the right-hand side - it is M + m. That is the mass of the monkey plus the mass of that skateboard, which is nine plus one which is ten. So, let's use a neutral color for that. So, ten kilograms, the
total mass, times v, which we need to calculate. ... And look, we can now calculate what v is, doing some algebra. So, the ten divides out, and the kilogram cancels, and we are done. V will equal nine, and the units left out
is meters per second, and that's our answer. So, once the monkey
jumps on that skateboard, the board takes off with a
speed of nine meters per second. All right, let's solve one more. Can you pause and try
and solve this yourself? Do it the same way. First, figure out what is given. Maybe draw a diagram,
collect all that data, and then see if we can somehow solve this problem. All right, let's see what's given. We have a two kilogram toy truck moving at four meters per second is about to collide with a one kilogram toy car moving at 17 meters per second in the opposite direction. Find their combined
velocity after collision, if they stick to each other. So, we have two toys coming
in the opposite direction, so here is the diagram. And so, we have a toy truck and a car moving towards each other, and then they collide
and stick to each other, which means after collision,
they will move together, and we need to figure out with what speed and direction will they
be moving together? So, we know their masses. We are given the truck
weighs two kilograms, and the car has a mass of one kilogram. We also know their initial velocities. The truck is coming in at four, and the car is coming in at 17 in the opposite direction, and we need to find what their combined velocity is after the collision. Okay. Where do we start? Again, because this is
a collision problem, we can start the same way we did before. We can see the total momentum of the car and the truck before collision should equal the total momentum after the collision, and again, just like before, because both of them have the same final velocity, we have pulled that v
out from that equation. Now, all we need to do is plug in these values and calculate what v is. So, if you couldn't do this before, can you pause this one more time, and see if you can do it from here, and, remember, these two are coming in the opposite direction, so please take care of that. Again, give it a try. All right, let's substitute. So, the mass of the truck is two kilogram into the initial velocity of that truck is four meters per second, plus this time this object is not at rest; it's moving. It's mass is one kilogram, and now comes the important part. Can we say it's velocity
is 17 meters per second? No, we can't. The main reason is velocities
are direction sensitive. So, if this velocity is to the right, and we are taking that as positive, this velocity is in
the opposite direction. So, if this is positive direction, we should call this negative, and that's very important, otherwise, we get the wrong answer. So, this is negative. Okay, when things are in
the opposite direction, remember, one velocity is positive. One is negative. And that should equal their total mass which is M + m, that
is, just three kilograms into v... And now if we simplify
this particular equation, which I'm pretty sure you can do. So, I leave that thing
to you just to save time. We will get V equals (if we simplify that) minus three meters per second And, again, you can pause and just verify that you get that answer, and that's our answer. So, this means after collision, they start moving at
three meters per second, but what does that minus sign mean? Well, it's telling us about the direction. You see, since we took this as positive, and we took this as negative, that means we chose
the right side velocity as positive and the left
side velocity as negative. That's what we did, and since we're getting a negative answer, this now means that this whole thing will start moving to the left. If we had gotten a positive answer, it would have meant the whole thing was going to the right, okay? So, the final answer not
only tells us the speed, it also tells us in which direction the whole thing is going, and so whenever we have
problems with collisions, we can always start with
momentum conservation.