a 300kg motorboat is turned off as it approaches a dock and coasts towards it at .5 m/s. Isaac, whose mass is 62 kg jumps off the front of the boat with a speed of 3m/s relative to the boat. what is the velocity of the boat after Isaac jumps?.

-- Before he jumps, the mass of (Isaac + boat) = (300 + 62) = 362 kg,their speed toward the dock is 0.5 m/s, and their linear momentum is  Momentum = (mass) x (speed) = (362kg x 0.5m/s) = 181 kg-m/srelative to the dock. So this is the frame in which we'll need to conservemomentum after his dramatic leap.After the jump:-- Just as Isaac is coiling his muscles and psyching himself up for the jump,he's still moving at 0.5 m/s toward the dock.  A split second later, he has leftthe boat, and is flying through the air at a speed of 3 m/s relative to the boat.That's 3.5 m/s relative to the dock.    His momentum relative to the dock is (62 x 3.5) = 217 kg-m/s toward it.But there was only 181 kg-m/s total momentum before the jump, and Isaactook away 217 of it in the direction of the dock.  The boat must now provide(217 - 181) = 36 kg-m/s of momentum in the opposite direction, in order to keep the total momentum constant.Without Isaac, the boat's mass is 300 kg, so                       (300 x speed) = 36 kg-m/s .Divide each side by 300:  speed = 36/300 = 0.12 m/s , away from the dock.=======================================Another way to do it . . . maybe easier . . . in the frame of the boat.In the frame of the boat, before the jump, Isaac is not moving, so nobody and nothing has any momentum.  The total momentum ofthe boat-centered frame is zero, which needs to be conserved.Isaac jumps out at 3 m/s, giving himself (62 x 3) = 186 kg-m/s of momentum in the direction toward the dock.Since 186 kg-m/s in that direction suddenly appeared out of nowhere,there must be 186 kg-m/s in the other direction too, in order to keep the total momentum zero. In the frame of measurements from the boat, the boat itself must startmoving in the direction opposite Isaac's jump, at just the right speed  so that its momentum in that direction is 186 kg-m/s.The mass of the boat is 300 kg so                                                         (300 x speed) = 186Divide each side by 300:  speed = 186/300 = 0.62 m/s    away from the jump.Is this the same answer as I got when I was in the frame of the dock ?I'm glad you asked. It sure doesn't look like it.The boat is moving 0.62 m/s away from the jump-off point, and away from the dock.To somebody standing on the dock, the whole boat, with its intrepid passengerand its frame of reference, were initially moving toward the dock at 0.5 m/s.Start moving backwards away from that at 0.62 m/s, and the person standingon the dock sees you start to move away from him at 0.12 m/s, and that's thesame answer that I got earlier, in the frame of reference tied to the dock.  yay !By the way ... thanks for the 6 points.  The warm cloudy water and crusty green bread are delicious.