 We're now going to take a look at a special case of the conservation of linear momentum and there are a couple of them that we'll look at. The first one will be with the control volume moving at a constant velocity and then the second special case that we'll look at is where the control volume is accelerating. But let's begin with the first of those two. So the control volume moving at constant velocity. Okay so what we find is that when we looked at conservation of linear momentum in previous lectures we were dealing with situations where we had non-accelerating and for the most part the control volume was always fixed. So what we're going to do we're going to take a look at a case where first of all the control volume is moving. So if we have a control volume that's moving let me draw out a schematic and what we're going to do we're going to have an inertial reference frame. So this will be what we refer to as being a fixed or stationary reference frame and for that I'm going to use capitals and then we will have a moving reference frame that the control volume is attached to. So what we have here we have an inertial reference frame and that is shown to the left. So this is our inertial reference frame shown over here and then we have a reference frame that is moving and within that moving reference frame that is where our control volume is located and the velocity at which the reference frame is moving is V RF. So that denotes the velocity of the reference frame. Now there are a couple of rules we need to abide or work by when we're dealing with the case of a constant velocity control volume. So the rules we need to abide by the first one is that all velocities that are coming into or leaving from the control volume are measured relative to the control volume. So we have to look at relative velocity and so if we have a velocity that is in the inertial frame we have to then convert that and look at the relative velocity with respect to the control volume and the second one is that all time derivatives are measured relative to the control volume as well. So those are some of the basic rules we have for a control volume that's moving at linear velocity. Now what we're going to do let's go back to the equation and this was Reynolds transport theorem but remember it's the equation that enables us to go from a system formulation which is all of our basic laws are expressed with a system formulation of fixed mass and it enabled us to go to the control volume formulation and we obtained this equation expressed in terms of n which is some extensive property that we would have be mass momentum energy or angular momentum. So this is an expression for the time rate of change of that extensive property and it is then expressed in terms of on the left it's for fixed mass and on the right is for the control volume formulation and you'll see one of the changes that we have here is we've expressed the velocity in this formulation here in terms of the velocity in our moving reference frame. So if we take a look at linear momentum remember that our extensive property was defined as being this and our intensive property then. So what we end up with for the case of linear momentum the force balance that we have from f equals ma we have surface forces plus body forces and then on the right hand side we have our time rate of change integrated across the control volume and our velocity here is measured with respect to the control volume. So when you look at this if you compare it back to what we had from before it looks almost identical the only thing that's changed is that we're writing velocity with respect to this moving control volume reference frame and and so that is the biggest change that we have in going that that's a case of constant linear velocity but in this equation v x y z in vector is the velocity with respect to the control volume. So that is the thing that you have to be careful with but if you do that you can then go ahead and solve problems involving linearly moving a constant velocity control volume and so what we'll do in the next segment is we're going to apply this to a problem and then we'll see how to work through control volume analysis when the control volume is moving at a constant velocity.