2 people,1 with 3 times the mass of the other, attempt a tug-of-war on frictionless ice. Show that the heavier person will gain a speed 1/3 that of the lighter person.
momentum before = 0*m + 0 * 3m = 0
momentum after =u *m + v*3m
momentum after = momentum before
um+3vm=0
vm = -um/3
v = -u/3
To show that the heavier person will gain a speed one-third that of the lighter person during a tug-of-war on frictionless ice, we need to consider the concept of conservation of momentum.
Conservation of momentum states that the total momentum of a closed system remains constant unless acted upon by external forces. In this case, the closed system consists of the two individuals participating in the tug-of-war.
Let's denote the mass of the lighter person as m1 and the mass of the heavier person as m2, with m2 being three times greater than m1. We'll also denote the initial velocities of the lighter and heavier individuals as v1 and v2, respectively. Since the surface is frictionless, there are no external forces acting on the system except for the force exerted by each individual on the other.
According to the principle of conservation of momentum:
Total initial momentum = Total final momentum
(m1)(v1) + (m2)(v2) = (m1)(v'1) + (m2)(v'2)
Where v'1 and v'2 represent the final velocities of the lighter and heavier individuals, respectively.
In this scenario, we can assume that the individuals start from rest, so v1 = v2 = 0.
The equation then simplifies to:
(m1)(0) + (m2)(0) = (m1)(v'1) + (m2)(v'2)
0 = (m1)(v'1) + (m2)(v'2)
Now, since the force exerted by each individual on the other is equal in magnitude but opposite in direction (Newton's third law of motion), we can say that the magnitudes of the final velocities will be equal, but the direction will be opposite:
|v'1| = |v'2|
Since m2 is three times greater than m1, we can rewrite the equation as:
0 = (m1)(v'1) + (3m1)(v'1)
0 = 4(m1)(v'1)
Simplifying further:
0 = v'1
This equation tells us that the lighter person will gain a speed of zero. However, since the magnitude of the velocity is the same for both individuals, the heavier person will gain a speed of zero as well.
Therefore, the conclusion is that the heavier person will not gain any speed during the tug-of-war, while the lighter person will also have a speed of zero.
Hence, the heavier person will not gain any speed, and the lighter person will also have a speed of zero.