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 or the woman can return to the boat, we need to analyze the conservation of momentum.
Firstly, let's calculate the initial momentum of the system. The initial momentum of the boat, man, and woman is given by:
Initial momentum = (mass of the boat) x (velocity of the boat) + (mass of the man) x (velocity of the man) + (mass of the woman) x (velocity of the woman)
Plugging in the values, we get:
Initial momentum = (60 kg) x (0 m/s) + (70 kg) x (0 m/s) + (50 kg) x (0 m/s)
= 0 kg m/s
Since the motor fails, the boat comes to a stop, so the velocity of the boat is zero. Therefore, the initial momentum of the entire system is zero.
Now, let's consider the man swimming back to the boat with a swimming speed of 1 m/s. When the man jumps back into the water, his momentum will change. The final momentum of the system will be:
Final momentum = (mass of the boat) x (velocity of the boat) + (mass of the man) x (velocity of the man) + (mass of the woman) x (velocity of the woman)
Plugging in the values, we get:
Final momentum = (60 kg) x (0 m/s) + (70 kg) x (-1 m/s) + (50 kg) x (0 m/s)
= -70 kg m/s
As the final momentum is not zero, it means that the man cannot swim back to the boat with a swimming speed of 1 m/s. His swimming speed is not sufficient to counteract the forward momentum of the system.
Now let's consider the woman diving off the boat at 3 m/s in the opposite direction. Similar to before, the final momentum of the system will be:
Final momentum = (mass of the boat) x (velocity of the boat) + (mass of the man) x (velocity of the man) + (mass of the woman) x (velocity of the woman)
Plugging in the values, we get:
Final momentum = (60 kg) x (0 m/s) + (70 kg) x (0 m/s) + (50 kg) x (-3 m/s)
= -150 kg m/s
As we can see, the final momentum is different from zero, indicating that the woman's action also cannot bring the boat to a complete stop.
Therefore, neither the man nor the woman can change the boat's motion enough to bring it to a complete stop and return to the boat with their respective swimming speeds.