Two ice skaters, each with a mass of 72.0 kg, are skating at 5.10 m/s when they collide and stick together. If the angle between their initial directions was 117°, determine the components of their combined velocity after the collision. (Let the initial motion of skater 1 be in the positive x direction and the initial motion of skater 2 be at an angle of 117° measured counterclockwise from the positive x-axis.)
Skater 1 is just 72*5.1 in x
Skater 2 is 72*5.1cos117 in x and 72*5.1sin117 in y
Sum x and y and using pythagorean & tan-1 find magnitude and direction of resultant
To determine the components of their combined velocity after the collision, we can use the principles of conservation of momentum and conservation of kinetic energy.
First, let's find the initial x and y components of velocity for each skater:
For Skater 1:
Initial velocity of Skater 1 in the x-direction (V1x) = 5.10 m/s
Initial velocity of Skater 1 in the y-direction (V1y) = 0 m/s
For Skater 2:
The angle between the initial direction of Skater 2 and the x-axis is 117° counterclockwise. We need to find the x and y components of Skater 2's initial velocity.
Initial velocity of Skater 2 in the x-direction (V2x) can be found using trigonometry:
V2x = magnitude of velocity * cos(angle)
V2x = 5.10 m/s * cos(117°)
V2x = -2.42 m/s (negative because it is to the left of the positive x-axis)
Initial velocity of Skater 2 in the y-direction (V2y) can be found using trigonometry:
V2y = magnitude of velocity * sin(angle)
V2y = 5.10 m/s * sin(117°)
V2y = 4.47 m/s
Next, let's find the total momentum in the x-direction (Px) and the total momentum in the y-direction (Py) before the collision:
Px = (mass of Skater 1 * V1x) + (mass of Skater 2 * V2x)
Px = (72.0 kg * 5.10 m/s) + (72.0 kg * -2.42 m/s)
Px = 0 kg·m/s
Py = (mass of Skater 1 * V1y) + (mass of Skater 2 * V2y)
Py = (72.0 kg * 0 m/s) + (72.0 kg * 4.47 m/s)
Py = 322.6 kg·m/s
Since momentum is conserved in both x and y directions, the total momentum after the collision will be equal to the total momentum before the collision. Therefore, total momentum after the collision in x-direction (P'x) will be 0 kg·m/s, and the total momentum after the collision in y-direction (P'y) will be 322.6 kg·m/s.
Now, let's find the combined velocity components in x and y directions after the collision (V'x and V'y):
For V'x:
Since there is no momentum in the x-direction after the collision, V'x = 0 m/s.
For V'y:
Using Py = (mass of Skater 1 + mass of Skater 2) * V'y:
322.6 kg·m/s = (72.0 kg + 72.0 kg) * V'y
322.6 kg·m/s = 144.0 kg * V'y
V'y = 322.6 kg·m/s / 144.0 kg
V'y = 2.24 m/s
Therefore, the combined velocity after the collision has a magnitude of 2.24 m/s in the y-direction and 0 m/s in the x-direction.
To find the components of their combined velocity after the collision, we can use the laws of conservation of momentum and conservation of kinetic energy.
1. First, let's find the initial momenta before the collision for each skater.
The initial momentum for skater 1 (m1) can be calculated as:
p1 = m1 * v1 = 72.0 kg * 5.10 m/s
p1 = 367.2 kg·m/s
The initial momentum for skater 2 (m2) can be calculated as:
p2 = m2 * v2 = 72.0 kg * 0 m/s (since skater 2 is initially at rest)
p2 = 0 kg·m/s
2. Next, let's find the components of the velocities of each skater using trigonometry.
The x-component of velocity for skater 1 (v1x) can be given by:
v1x = v1 * cos(angle) = 5.10 m/s * cos(0°) = 5.10 m/s
The y-component of velocity for skater 1 (v1y) can be given by:
v1y = v1 * sin(angle) = 5.10 m/s * sin(0°) = 0 m/s
The x-component of velocity for skater 2 (v2x) can be given by:
v2x = v2 * cos(angle) = 0 m/s * cos(117°) = 0 m/s
The y-component of velocity for skater 2 (v2y) can be given by:
v2y = v2 * sin(angle) = 0 m/s * sin(117°) = 0 m/s
3. Now, let's find the total momentum and total kinetic energy before the collision.
Total initial momentum before the collision:
p_total_initial = p1 + p2 = 367.2 kg·m/s + 0 kg·m/s = 367.2 kg·m/s
Total initial kinetic energy before the collision:
KE_total_initial = 1/2 * m1 * v1^2 + 1/2 * m2 * v2^2
KE_total_initial = 1/2 * 72.0 kg * (5.10 m/s)^2 + 1/2 * 72.0 kg * (0 m/s)^2
KE_total_initial = 876.24 J + 0 J = 876.24 J
4. After the collision, the two skaters stick together, so their combined mass is now 2 * 72.0 kg = 144.0 kg.
5. Let's define the combined velocity of the two skaters as Vf.
The x-component of the combined velocity (Vfx) can be found using the conservation of momentum:
Vfx = (p_total_initial) / (m_total_final)
Vfx = 367.2 kg·m/s / 144.0 kg
Vfx ≈ 2.55 m/s
Since the skaters stick together after the collision, Vfx will be the same for both skaters.
The y-component of the combined velocity (Vfy) is equal to the sum of the y-components of the individual velocities before the collision:
Vfy = v1y + v2y = 0 m/s + 0 m/s = 0 m/s
Therefore, the components of their combined velocity after the collision are:
Vfx ≈ 2.55 m/s (in the x-direction)
Vfy = 0 m/s (in the y-direction)