Two 75.0-kg hockey players skating at 5.75 m/s collide and stick together. If the angle between their initial directions was 135°, what is their speed after the collision? (Let the motion of player 1 be in the positive x-direction and the motion of player 2 be at an angle of 135° measured counterclockwise from the positive x-axis.)
To find their speed after the collision, we will use the principles of conservation of momentum and conservation of kinetic energy.
First, let's find the initial momentum of each player. The momentum of an object is given by the product of its mass and velocity.
Player 1:
Mass (m1) = 75.0 kg
Velocity (v1) = 5.75 m/s
Player 2:
Mass (m2) = 75.0 kg
Velocity (v2) = 5.75 m/s
To find the initial x-components of their velocities, we can use trigonometry.
The x-component of Player 1's velocity (v1x) is given by:
v1x = v1 * cos(angle)
v1x = 5.75 m/s * cos(0°) = 5.75 m/s
The x-component of Player 2's velocity (v2x) is given by:
v2x = v2 * cos(angle)
v2x = 5.75 m/s * cos(135°)
To find the initial y-components of their velocities:
The y-component of Player 1's velocity (v1y) is given by:
v1y = v1 * sin(angle)
v1y = 5.75 m/s * sin(0°) = 0 m/s
The y-component of Player 2's velocity (v2y) is given by:
v2y = v2 * sin(angle)
v2y = 5.75 m/s * sin(135°)
Now let's find the resultant momentum (x-component) of the system before the collision:
P_initial_x = m1 * v1x + m2 * v2x
Similarly, let's find the resultant momentum (y-component) of the system before the collision:
P_initial_y = m1 * v1y + m2 * v2y
Since the players stick together after the collision, their final momentum will be the same, but the velocities might change.
The combined mass after sticking together = m1 + m2.
Let's assume their final velocity (v_f) after the collision.
To find the final momentum (x-component) of the system:
P_final_x = (m1 + m2) * v_f
Similarly, let's find the final momentum (y-component) of the system:
P_final_y = 0
Since momentum is conserved in both x and y directions, we can write:
P_initial_x = P_final_x,
P_initial_y = P_final_y.
So, we have:
m1 * v1x + m2 * v2x = (m1 + m2) * v_f,
m1 * v1y + m2 * v2y = 0.
Now, we can solve these equations to find the value of v_f.
Substituting the values we know:
75.0 kg * 5.75 m/s + 75.0 kg * 5.75 m/s * cos(135°) = (75.0 kg + 75.0 kg) * v_f.
Calculate the values on the left side of the equation, and then solve for v_f.
Once you have found the value of v_f, you will have the final speed after the collision between the two hockey players.