A 91.76-kg boater, initially at rest in a stationary 62.54-kg canoe, steps out of the canoe and onto the dock. If the boater moves out of the boat with a velocity of 3.52 m/s to the right, what is the final velocity of the boat?
To find the final velocity of the boat, we can apply the concept of conservation of momentum.
The law of conservation of momentum states that the total momentum before an event is equal to the total momentum after the event, as long as there are no external forces acting on the system.
The momentum of an object is defined as the product of its mass and velocity. Mathematically, momentum (p) is given by p = m * v, where m is the mass and v is the velocity.
In this scenario, the initial momentum of the system (boat + boater) is zero because both the boat and boater are at rest.
Initially:
Momentum of boat = 0 kg*m/s
Momentum of boater = 0 kg*m/s
When the boater steps out of the canoe onto the dock, the boat experiences an equal and opposite reaction due to the conservation of momentum.
Final momentum of boat = - (mass of boater) * (velocity of boater)
Let's calculate the final momentum of the boat:
Final momentum of boat = - (91.76 kg) * (3.52 m/s)
Final momentum of boat = - 322.35 kg*m/s
Since the boat initially had zero momentum, the final momentum of the boat will be opposite in direction to the momentum of the boater.
Therefore, the final velocity of the boat can be determined by dividing the final momentum of the boat by the mass of the boat:
Final velocity of boat = (Final momentum of boat) / (mass of boat)
Final velocity of boat = (-322.35 kg*m/s) / (62.54 kg)
Final velocity of boat = -5.15 m/s
Hence, the final velocity of the boat is -5.15 m/s. The negative sign indicates that the boat is moving in the opposite direction to that of the boater.
To find the final velocity of the boat, we can use the law of conservation of momentum. According to this law, the total momentum before the boater steps out of the canoe must be equal to the total momentum after the boater steps out.
The momentum of an object is equal to its mass multiplied by its velocity. Therefore, we can calculate the initial and final momenta as follows:
Initial momentum of boater + Initial momentum of canoe = Final momentum of boater + Final momentum of canoe
The initial momentum of the boater is given by:
Initial momentum of boater = mass of boater * initial velocity of boater
Substituting the given values:
Initial momentum of boater = 91.76 kg * 3.52 m/s = 322.35 kg·m/s
The initial momentum of the canoe is given by:
Initial momentum of canoe = mass of canoe * initial velocity of canoe
Given that the canoe is initially at rest, the initial velocity of the canoe is 0, so the initial momentum of the canoe is 0. (since momentum of an object at rest is 0)
Therefore, the equation becomes:
322.35 kg·m/s + 0 = Final momentum of boater + Final momentum of canoe
Let's denote the final velocity of the canoe as Vf. The final momentum of the boater is then given by:
Final momentum of boater = mass of boater * final velocity of boater
Substituting the given values:
Final momentum of boater = 91.76 kg * 3.52 m/s = 322.35 kg·m/s
The final momentum of the canoe is given by:
Final momentum of canoe = mass of canoe * final velocity of canoe
Substituting the given values:
Final momentum of canoe = 62.54 kg * Vf
By substituting all the values in the conservation of momentum equation, we get:
322.35 kg·m/s + 0 = 322.35 kg·m/s + 62.54 kg * Vf
To solve for Vf, we can simplify the equation:
0 = 62.54 kg * Vf
Dividing both sides of the equation by 62.54 kg, we get:
Vf = 0
Therefore, the final velocity of the canoe is 0 m/s, meaning it remains at rest.