19. Two people of unequal mass are initially standing still on the ice with negligible friction. They then simultaneously push each other horizontally. Afterward, which of the following is true? *

-The center of mass of the two person system moves in the direction of the less massive person.
-The momenta of the people are of equal magnitude.
-The less massive person has a smaller initial acceleration than the more massive person.
-The kinetic energies of the two people are equal.
-The speeds of the two people are equal.

To determine which of the statements is true, let's analyze the scenario using the principles of physics.

When two people of unequal mass push each other horizontally on ice, they experience an action-reaction pair of forces according to Newton's third law. According to this law, the force exerted by one person on the other is equal in magnitude but opposite in direction to the force exerted by the second person on the first.

Now, let's examine each statement:

1. The center of mass of the two person system moves in the direction of the less massive person.
The center of mass of a system is a point that represents the average position of all the mass in that system. In this scenario, the center of mass will move in the direction of the net force acting on the system. Since the forces exerted by both individuals are equal in magnitude and opposite in direction, and the masses are unequal, the net force will be greater in the direction of the less massive person. Therefore, statement 1 is true.

2. The momenta of the people are of equal magnitude.
Momentum is defined as the product of an object's mass and velocity. When the two people push each other, their momenta change due to the action-reaction forces. Since the forces are equal in magnitude and opposite in direction, the change in momentum for each person will be equal and opposite. Therefore, statement 2 is true.

3. The less massive person has a smaller initial acceleration than the more massive person.
According to Newton's second law, F = ma, the acceleration of an object is inversely proportional to its mass for the same applied force. So, the less massive person will experience greater acceleration than the more massive person because the force exerted on them is equal. Therefore, statement 3 is false.

4. The kinetic energies of the two people are equal.
Kinetic energy is given by the equation KE = 1/2 mv^2, where m is the mass and v is the velocity of the object. Since the masses and velocities of the two people are unequal, their kinetic energies will also be unequal. Therefore, statement 4 is false.

5. The speeds of the two people are equal.
Speed is defined as the magnitude of velocity, disregarding the direction. In this scenario, the two people are pushing each other horizontally, so their velocities will have equal magnitudes due to the equal forces exerted on them. However, their masses are unequal, so their speeds will not be the same. Therefore, statement 5 is false.

In conclusion, statement 1 and statement 2 are true, while statement 3, statement 4, and statement 5 are false.

The correct answer is:

-The momenta of the people are of equal magnitude.

According to the law of conservation of momentum, the total momentum of an isolated system remains constant if no external forces act on it. In this case, since the two people are pushing each other, the net external force acting on the system is zero, and therefore the total momentum of the system remains constant.

As a result, the momenta of the two people must be equal in magnitude but opposite in direction. This means that if one person has a certain momentum, the other person will have the same magnitude of momentum but in the opposite direction.

The other statements mentioned are not true in this scenario.

the momenta is the same in both direction. Newton's third law.