Two skaters of mass m1=50kg and m2=70kg are standing motionless on a horizontal ice surface. They are initially a distance L=7.0 meters apart. They hold a massless rope between them. After pulling the rope, the skater of mass m1 has moved a distance l=2.0 meters away from his initial position. We can completely neglect friction in this problem.
What is the distance L' between the two skaters when the skater of mass m1 has moved a distance l? (in meters)
L'=
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To solve this problem, we can use the principle of conservation of momentum. No external forces act on the system of skaters and rope, so the total momentum before and after pulling the rope remains the same.
The initial total momentum is given by the sum of the individual momenta of each skater:
p_initial = m1 * v1_initial + m2 * v2_initial
Since both skaters are standing motionless, their initial velocities are zero. Hence, the initial total momentum is zero.
After pulling the rope, the skater of mass m1 moves a distance l = 2.0 meters away from his initial position. Let's assume that the skater of mass m2 also moves a distance L'.
Applying conservation of momentum, we can write:
p_final = m1 * v1_final + m2 * v2_final
The final momentum is still zero because the skaters are initially at rest. Assuming both skaters move in opposite directions, the final velocities can be written as follows:
v1_final = m2 / (m1 + m2) * v_relative
v2_final = -m1 / (m1 + m2) * v_relative
where v_relative is the relative velocity between the skaters. We need to find v_relative.
Since the skater of mass m1 moved a distance l = 2.0 meters away from his initial position, we can find the relative velocity using the time taken for this motion:
v_relative = l / t
To find the time taken, we can use the fact that the distance traveled by an object is equal to its average velocity multiplied by the time taken:
l = (v1_initial + v1_final) / 2 * t
Since both skaters start from rest and their average velocities are zero, we can simplify the equation to:
l = v1_final * t
Rearranging this equation, we find:
t = l / v1_final
Substituting the values:
t = 2.0 / (m2 / (m1 + m2) * v_relative)
Now, we can substitute the values of t and v_relative into the equations for v1_final and v2_final, and calculate their values.
Finally, we can substitute the values of v1_final and v2_final into the equation for the final total momentum, set it equal to zero, and solve for L'.
L' is the distance between the skaters when the skater of mass m1 has moved a distance l.