To get off a frozen frictionless lake, a 70.0kg person takes off a 0.150-kg shoe and throws it horizontally directly away from the shore with a speed of 2.00m/s. If the person is 5.0 m from the shore, how long does it take for him to reach it?
To find out how long it takes for the person to reach the shore, we need to consider the conservation of momentum.
First, let's calculate the initial momentum of the system before the person throws the shoe. The initial momentum is given by the sum of the person's momentum and the shoe's momentum:
Initial momentum = Person's momentum + Shoe's momentum
The person's momentum can be calculated using the formula:
Person's momentum = Person's mass × Person's velocity
The shoe's momentum is given by:
Shoe's momentum = Shoe's mass × Shoe's velocity
Let's calculate the momentum values:
Person's momentum = 70.0 kg × 0 m/s (since the person is at rest initially)
Shoe's momentum = 0.150 kg × 2.00 m/s (since the person throws the shoe away with a velocity of 2.00 m/s)
Now we can find the initial momentum:
Initial momentum = Person's momentum + Shoe's momentum
Next, let's consider the final momentum of the system. Since the person is 5.0 m from the shore and wants to reach it, we can assume that the person throws the shoe right as they start moving towards the shore, and that the shoe does not have any effect on their motion. Therefore, the final momentum is simply the person's momentum:
Final momentum = Person's momentum
Using the principle of conservation of momentum, we can equate the initial momentum and the final momentum:
Initial momentum = Final momentum
Solving for the final momentum, which is the person's momentum:
Person's momentum = Initial momentum
Now that we have found the person's momentum, we can find the person's velocity using the formula:
Person's momentum = Person's mass × Person's velocity
Now we can solve for the person's velocity:
Person's velocity = Person's momentum / Person's mass
Once we have the person's velocity, we can calculate the time it takes for the person to reach the shore using the formula:
Time = Distance / Velocity
Using the given values, we can substitute them into the formulas and calculate the time it takes for the person to reach the shore.