A 70-kg man and a 50-kg woman are in a 60-kg boat when its motor fails. The man dives into the water with a horizontal speed of 3m/s in order to swim ashore. If he changes his mind, can he swim back to the boat if his swimming speed is 1m/s? If not, can the woman change the boat's motion enough by diving off it at 3 m/s in the opposite direction? Could she then return to the boat herself if her swimming speed is also 1 m/s?

To determine whether the man can swim back to the boat, we first need to consider the initial momentum of the system. The initial momentum is given by the sum of the momentum of the man, woman, and boat.

Initial momentum = (mass of man × velocity of man) + (mass of woman × velocity of woman) + (mass of boat × velocity of boat)

= (70 kg × 3 m/s) + (50 kg × 0 m/s) + (60 kg × 0 m/s)
= 210 kg·m/s

Now, let's consider the situation after the man dives into the water and starts swimming back to the boat. His momentum in the opposite direction will be:

Final momentum of man = (mass of man × swimming speed)

= 70 kg × 1 m/s
= 70 kg·m/s

At this point, we can observe that the man's momentum is not enough to compensate for the initial momentum of the system. Therefore, the man cannot swim back to the boat on his own.

Now, let's move on to the woman's scenario. When the woman jumps off the boat with a velocity of 3 m/s in the opposite direction, her momentum in the opposite direction will be:

Final momentum of woman = (mass of woman × diving velocity)

= 50 kg × (-3 m/s) (negative sign indicates opposite direction)
= -150 kg·m/s

Now, let's consider the situation when the woman starts swimming back to the boat with a swimming speed of 1 m/s. Her momentum in the opposite direction will be:

Final momentum of woman = (mass of woman × swimming speed)

= 50 kg × (-1 m/s)
= -50 kg·m/s

The total momentum after the woman jumps off and starts swimming will be the sum of her momentum and the man's momentum:

Total final momentum = Final momentum of woman + Final momentum of man

= (-150 kg·m/s) + (70 kg·m/s)
= -80 kg·m/s

Since the total final momentum is negative, it means the boat has the opposite direction of motion, and the woman cannot swim back to the boat on her own either.

In conclusion, neither the man nor the woman can swim back to the boat on their own under the given conditions.