Two 72.0-kg hockey players skating at 7.00 m/s collide and stick together. If the angle between their initial directions was 130 degrees , what is their speed after the collision?
To find the speed of the two hockey players after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity. In this case, both players have the same mass (72.0 kg), but they are moving in different directions.
Let's first calculate the initial momentum of each player:
Player 1:
Mass (m1) = 72.0 kg
Velocity (v1) = 7.00 m/s
Momentum (p1) = m1 * v1
Player 2:
Mass (m2) = 72.0 kg
Velocity (v2) = 7.00 m/s
Momentum (p2) = m2 * v2
Next, let's calculate the total initial momentum:
Total initial momentum = p1 + p2
Now, since the players stick together after the collision, their combined mass will be twice their individual masses:
Total mass after the collision = 2 * (mass of each player)
Let's calculate the total mass after the collision:
Total mass after the collision = 2 * 72.0 kg
Finally, since the total momentum after the collision should be equal to the total initial momentum, we can set up an equation:
Total initial momentum = Total mass after the collision * final velocity
Let's solve this equation for the final velocity:
Total initial momentum = Total mass after the collision * final velocity
p1 + p2 = 2 * 72.0 kg * final velocity
Now substitute the values of p1 and p2:
(m1 * v1) + (m2 * v2) = 2 * 72.0 kg * final velocity
Substituting the given values:
(72.0 kg * 7.00 m/s) + (72.0 kg * 7.00 m/s) = 2 * 72.0 kg * final velocity
Finally, solve for the final velocity:
(2 * 72.0 kg * 7.00 m/s) / (2 * 72.0 kg) = final velocity
The mass on both sides will cancel out, leaving us with:
7.00 m/s = final velocity
Therefore, the speed of the two hockey players after the collision is 7.00 m/s.
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
Conservation of momentum states that the total momentum of a system remains constant before and after a collision, assuming no external forces act on the system. Mathematically, this can be expressed as:
Total initial momentum = Total final momentum
Momenta of the two players before collision:
Player 1 momentum = mass of player 1 * velocity of player 1
Player 2 momentum = mass of player 2 * velocity of player 2
Since they stick together after the collision, their final mass is the sum of their initial masses, and their final velocity is the velocity of the center of mass of the system. Let's denote the final mass as Mf and the final velocity as Vf.
After the collision, the total momentum becomes:
Total final momentum = (Mf) * (Vf)
Conservation of kinetic energy states that the total kinetic energy of a system remains constant before and after a collision if no external forces act on the system. Mathematically, this can be expressed as:
Total initial kinetic energy = Total final kinetic energy
Initial kinetic energy of player 1 = (1/2) * mass of player 1 * (velocity of player 1)^2
Initial kinetic energy of player 2 = (1/2) * mass of player 2 * (velocity of player 2)^2
Final kinetic energy = (1/2) * (Mf) * (Vf)^2
Now, we can solve these equations to find the final velocity Vf.
Since the problem provides the mass (72 kg) and initial velocity (7.00 m/s) for both players, we can substitute these values into the equations. Before doing that, it's important to convert the angle between their initial directions (130 degrees) to radians, as it will be needed to calculate the final velocity:
Angle in radians = angle in degrees * (pi/180)
Angle in radians = 130 * (pi/180) = 2.268 radians
Now, we can substitute the given values into the equations:
Player 1 momentum = (72.0 kg) * (7.00 m/s) * cos(2.268 radians)
Player 2 momentum = (72.0 kg) * (7.00 m/s) * cos((180 - 2.268) radians)
Total initial momentum = Player 1 momentum + Player 2 momentum
Total final momentum = (Mf) * (Vf)
Total initial kinetic energy = Initial kinetic energy of player 1 + Initial kinetic energy of player 2
Total final kinetic energy = (1/2) * (Mf) * (Vf)^2
Finally, we can solve the equations to find the final velocity Vf.