A 125 kg flatbottomed boat is floating along at 5.0 m/s, headed east. Sally Sue, standing at one
end of the boat, starts to walk at 2.0 m/s, also to the east. If Sally’s mass is 65 kg, what is the
speed of the boat as she walks along?
To solve this problem, we can use the principle of conservation of momentum. The total momentum before Sally starts walking should be equal to the total momentum after she starts walking.
The momentum of an object is given by the product of its mass and velocity. In this case, we have two objects: the boat and Sally Sue.
Before Sally starts walking, the boat is moving eastward with a velocity of 5.0 m/s. We can calculate the momentum of the boat using the formula:
Momentum of the boat before = mass of the boat × velocity of the boat
= 125 kg × 5.0 m/s
= 625 kg·m/s eastward
Sally Sue has a mass of 65 kg and initially has zero velocity. So her momentum before she starts walking is zero:
Momentum of Sally Sue before = mass of Sally Sue × initial velocity of Sally Sue
= 65 kg × 0 m/s
= 0 kg·m/s eastward
After Sally starts walking at 2.0 m/s, we need to calculate the total momentum of both the boat and Sally Sue:
Total momentum after = momentum of the boat after + momentum of Sally Sue after
The momentum of the boat after Sally starts walking will be its mass times its final velocity. Since the boat continues to move with the same speed and direction, the velocity of the boat remains 5.0 m/s:
Momentum of the boat after = mass of the boat × velocity of the boat
= 125 kg × 5.0 m/s
= 625 kg·m/s eastward
The momentum of Sally Sue after she starts walking will be her mass times her velocity:
Momentum of Sally Sue after = mass of Sally Sue × velocity of Sally Sue
= 65 kg × 2.0 m/s
= 130 kg·m/s eastward
Now we can substitute these values back into the equation for total momentum after Sally starts walking:
Total momentum after = 625 kg·m/s eastward + 130 kg·m/s eastward
= 755 kg·m/s eastward
Since momentum is conserved, the total momentum before Sally starts walking must equal the total momentum after she starts walking. Therefore:
Total momentum before = 755 kg·m/s eastward
To find the resulting velocity of the boat after Sally starts walking, we divide the total momentum by the total mass of the boat and Sally Sue:
Total momentum before = total mass × resulting velocity
755 kg·m/s = (mass of the boat + mass of Sally Sue) × resulting velocity
Solving for the resulting velocity:
resulting velocity = 755 kg·m/s ÷ (mass of the boat + mass of Sally Sue)
The total mass is the sum of the masses of the boat and Sally Sue:
total mass = mass of the boat + mass of Sally Sue
Plug in the given values:
total mass = 125 kg + 65 kg
= 190 kg
Now substituting the values back into the equation for resulting velocity:
resulting velocity = 755 kg·m/s ÷ 190 kg
Finally, calculating the resulting velocity:
resulting velocity = 3.97 m/s eastward
Therefore, the speed of the boat as Sally walks along is approximately 3.97 m/s.