Two groups of canoeists meet in the middle of a lake. After a brief visit, a person in canoe 1 pushes on canoe 2 to separate the canoes. Suppose the speeds of the two canoes after they are pushed apart are 0.65 m/s for canoe 1 and 0.47 m/s for canoe 2. If the mass of canoe 1 is 270 kg , what is the mass of canoe 2?
To solve this problem, we can use the conservation of momentum principle. According to this principle, the total momentum before the interaction between the canoes is equal to the total momentum after the interaction.
The momentum of an object is defined as the product of its mass and velocity. Mathematically, momentum (p) is calculated using the formula:
p = m * v
Where:
p = momentum
m = mass of the object
v = velocity of the object
In this scenario, let's assume that the initial velocity of canoe 1 is v1, the initial velocity of canoe 2 is v2, and the final velocities after the push are v1' (0.65 m/s) and v2' (0.47 m/s).
According to the conservation of momentum principle, the total momentum before pushing is equal to the total momentum after pushing:
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
We are given the mass of canoe 1 (m1 = 270 kg) and the velocities after the push (v1' = 0.65 m/s and v2' = 0.47 m/s). We need to find the mass of canoe 2 (m2).
Let's rearrange the equation to solve for m2:
(m1 * v1) + (m2 * v2) - (m1 * v1') = m2 * v2'
Now substitute the given values:
(270 kg * v1) + (m2 * v2) - (270 kg * 0.65 m/s) = m2 * 0.47 m/s
From this equation, we can solve for m2:
270 kg * v1 + m2 * v2 - 270 kg * 0.65 m/s = m2 * 0.47 m/s
Let's substitute the given values for v1 and v2:
270 kg * 0.65 m/s + m2 * v2 - 270 kg * 0.65 m/s = m2 * 0.47 m/s
Simplifying the equation:
175.5 kg·m/s + m2 * v2 - 175.5 kg·m/s = m2 * 0.47 m/s
The units of kg·m/s cancel out:
m2 * v2 - m2 * 0.47 m/s = 0
Factor out m2 on the left side:
m2 * (v2 - 0.47 m/s) = 0
Since the product of m2 and (v2 - 0.47 m/s) is zero, either m2 = 0 or v2 - 0.47 m/s = 0.
m2 = 0 is not a valid solution since it means there is no mass for canoe 2. Therefore, we can conclude that:
v2 - 0.47 m/s = 0
Solving this equation for v2:
v2 = 0.47 m/s
So, the mass of canoe 2 is not dependent on the values provided in the problem statement. The mass could be any value.