period of problem 159 chapter 1. In this problem there is a raft on the surface of water. The mass of raft is m and there is a man mid-mass m on the raft who is moving with a velocity U dash in reference to the raft and travels a distance L dash and stops. Now we have to find out what is the velocity of raft, how much distance raft travels during this time and what is the force exerted on the raft by the man when he is moving. In such problems keep two three things in mind. This is based on conservation of momentum, conservation of momentum, and when you are applying conservation of momentum always take the absolute velocities. Means velocity with reference to ground stationary frame do not take the relative velocity with reference to a moving frame. Velocity with reference to ground number one, number two, avoid taking sign, avoid taking sign yourself. Means positive or negative let it come in solution to the problem, let it come. It is always better. Now in this case man is moving we man with respect to ground we want to know. We know man's velocity with respect to raft that is not enough. We man with respect to ground would be equal to we man with respect to raft plus we raft with respect to ground. Please remember this convention you will notice that these two will match. Means the suffix of this will match the prefix of this and these two will be same as this. Take the second part here you write the relative to this, take the second part on which this is moving. On the raft man was moving. So take the object which is moving and on which another object is moving in this sequence. Means one object moving on another object. This is first object, this is second object moving like this. So first take velocity of one with reference to two and then take velocity of two reference to ground. Follow this sequence and you will not commit any mistake. Write like this. Now for conservation of momentum there is no external force which is acting on the system as a whole. No force is acting and initially it is at rest. So the initial momentum is zero. What is the final momentum? Final momentum is mass of man velocity of man with reference to ground plus mass of raft velocity of raft with reference to ground. Do like this. Now write this velocity of mass with respect to ground. Man with respect to ground can be written from here. We man with respect to raft plus we raft with respect to ground plus m v r with respect to ground. So it becomes m v m with respect to raft plus v raft with respect to ground into m plus m is zero. So we get v raft with respect to ground is equal to minus m v man with respect to raft upon m plus m. This is the velocity of the raft with respect to ground and v man with respect to raft was given. So it was minus u t is given upon m plus m. This is the velocity. Now what is the distance travelled? What is the time taken by this person in moving distance L dash? We know u t is equal to L dash upon time. So time is L dash upon u t. So to find the distance travelled by raft multiply this by time t and you will get the distance. Sign means it is moving in opposite direction. The man's motion was in this direction and we had taken that as positive. If you are getting negative means it is moving in this direction. Now multiply this velocity by this it is L dash upon u t. This gets cancelled. So you get distance is equal to m L dash upon m plus m is the distance travelled by the raft. Now we have to find out the force exerted by a man on the raft. What is v raft with respect to ground is m minus m v man with respect to raft upon m plus m. This is the raft velocity. What is the momentum of raft? would be minus m is the mass of raft multiplied by this v m raft is nothing but u t is given m plus m. What is force is nothing but rate of change of momentum. The rate of change of momentum is nothing but the force exerted by a man because there is no other external force on the system. So force will be equal to dp upon dt is equal to minus m m d u upon dt divided by m plus m. So this is the force which is exerted by a man on the raft and that was the distance that solves the problem. Thank you.