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?
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 need to consider the conservation of momentum.
Let's first calculate the initial momentum of the system. The momentum is given by the mass multiplied by the velocity.
Initial momentum = (mass of the man + mass of the woman + mass of the boat) * 0 m/s
= (70 kg + 50 kg + 60 kg) * 0 m/s
= 180 kg * 0 m/s
= 0 kg·m/s
Now, if the man swims with a speed of 1 m/s, his momentum will be 70 kg * 1 m/s = 70 kg·m/s. However, since the boat and the woman are not moving at that point, the total momentum of the system will be 70 kg·m/s.
So, to swim back to the boat, the man needs to increase the momentum of the system to 0 kg·m/s. However, swimming with 1 m/s won't be sufficient to counter the initial momentum of 70 kg·m/s. Therefore, the man cannot swim back to the boat on his own.
Now, let's consider if the woman dives off the boat with a speed of 3 m/s in the opposite direction. This action would change the momentum of the system.
Momentum of the woman = mass of the woman * velocity of the woman
= 50 kg * (-3 m/s) (negative sign indicates opposite direction)
= -150 kg·m/s
The momentum of the boat and the man remains the same (since they are not moving).
Total momentum of the system after the woman dives off = Momentum of the boat + Momentum of the man + Momentum of the woman
= 0 kg·m/s + 0 kg·m/s + (-150 kg·m/s)
= -150 kg·m/s
Since the momentum of the system is now negative, the woman diving off the boat has indeed changed the momentum of the system. Therefore, the woman can change the boat's motion to a certain extent by diving off.
Finally, let's consider if the woman can swim back to the boat with her swimming speed of 1 m/s. Initially, she had a momentum of -150 kg·m/s when she dived off the boat. In order to swim back and return to the boat, she needs to increase her momentum to 0 kg·m/s.
However, swimming with 1 m/s won't be enough to increase her momentum by 150 kg·m/s. Therefore, 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 with their respective swimming speeds of 1 m/s.