Two skaters with masses of 66kg and 47kg , respectively, stand 5.5m apart, each holding one end of a piece of rope.
Part A
If they pull themselves along the rope until they meet, how far does each skater travel? (Neglect friction.)
Part B
If only the 47kg skater pulls along the rope until she meets her friend (who just holds onto the rope), how far does each skater travel?
Express your answers using two significant figures separated by a comma.
x66kg, x47kg
To solve Part A, we need to consider the conservation of momentum. Since the skaters are initially at rest, the total momentum before they start moving is zero. After they meet, their final momentum is also zero since they come to a stop. This means that the total change in momentum during this process is zero.
We can use the equation for momentum:
Mass × Velocity = Momentum
Let's assume that the 66kg skater travels a distance of x meters towards the 47kg skater.
For the 66kg skater:
Mass × Velocity = Momentum
66kg × v1 = 66kg × v1 (Initial) + 47kg × v2 (Final)
For the 47kg skater:
Mass × Velocity = Momentum
47kg × v2 = 66kg × v1 (Initial) + 47kg × v2 (Final)
Simplifying these equations, we have:
66v1 = 66v1 + 47v2
47v2 = 66v1 + 47v2
Rearranging the equations, we get:
66v1 - 66v1 = 47v2
-19v1 = 47v2
Now, let's substitute the known values:
-19x = 47(5.5 - x)
Solving this equation will give us the value of x, which represents the distance the 66kg skater travels.
To solve Part B, we only need to find the distance traveled by the 47kg skater when she pulls along the rope until she meets her friend. In this case, the 66kg skater is stationary.
Using the same equation as before, but setting the initial velocity of the 66kg skater to zero, we have:
0 = 47v2
Simplifying this equation gives us the value of v2, which represents the distance the 47kg skater travels.
By solving these equations, we can find the distances traveled by each skater.
Part A:
To find out how far each skater travels when they both pull themselves along the rope until they meet, we can start by calculating the total distance covered.
The skaters are initially 5.5m apart, and when they meet, they will be at the same location. So, the total distance covered is equal to the initial separation between them.
Therefore, both skaters will travel a distance of 5.5m.
Answer for Part A: x66kg = 5.5m, x47kg = 5.5m
Part B:
In this scenario, only the 47kg skater pulls along the rope until she meets her friend, who just holds onto the rope. To find out how far each skater travels, we can analyze their individual movements.
The 47kg skater is the only one moving, so she will cover the entire distance of 5.5m. Thus, the 47kg skater will travel a distance of 5.5m.
On the other hand, the 66kg skater is stationary, so he will not travel any distance. In this case, we can say that the 66kg skater traveled a distance of 0m.
Answer for Part B: x66kg = 0m, x47kg = 5.5m