Two skaters collide an embrace, in a completely inelastic collision. That is, they stick together after impact. The origin is placed at the point of collision. Alfred, whose mass is 83 kg, is originally moving east with speed 6.2 km/h. Barbara, whose mass is 55 kg, is originally moving north with speed 7.8 km/h. What is the velocity v of the couple after impact? Hint: You must find vx, vy and the angle theta.
V = 6.2 + 7.8i = 9.96km/h[51.5o].
To find the velocity v of the couple after impact, we need to find the x-component (vx), y-component (vy), and the angle (θ) of the velocity.
Step 1: Convert the given speeds to SI units (m/s).
Alfred's speed: 6.2 km/h = (6.2 * 1000) / 3600 m/s ≈ 1.72 m/s
Barbara's speed: 7.8 km/h = (7.8 * 1000) / 3600 m/s ≈ 2.17 m/s
Step 2: Use the conservation of momentum to find the x-component (vx).
In an inelastic collision, the total momentum before the collision is equal to the total momentum after the collision.
Here, the x-component of the momentum before the collision is only due to Alfred's motion and is given by:
P_x_before = mass_Alfred * velocity_Alfred_x
Since there is no initial x-component of momentum due to Barbara's motion, the x-component of momentum after the collision will be:
P_x_after = mass_Alfred_Barbara * velocity_x
Since momentum is conserved, we have:
P_x_before = P_x_after
mass_Alfred * velocity_Alfred_x = (mass_Alfred + mass_Barbara) * velocity_x
Plugging in the known values:
(83 kg) * (1.72 m/s) = (83 kg + 55 kg) * velocity_x
Simplifying the equation:
(83 kg) * (1.72 m/s) = (138 kg) * velocity_x
velocity_x ≈ (83 kg * 1.72 m/s) / (138 kg) ≈ 1.03 m/s
Thus, the x-component of the velocity (vx) after the collision is approximately 1.03 m/s.
Step 3: Use the conservation of momentum to find the y-component (vy).
Similar to the x-component, the y-component of the momentum before the collision is due to Barbara's motion and is given by:
P_y_before = mass_Barbara * velocity_Barbara_y
Since there is no initial y-component of momentum due to Alfred's motion, the y-component of momentum after the collision will be:
P_y_after = mass_Alfred_Barbara * velocity_y
Applying the conservation of momentum:
P_y_before = P_y_after
mass_Barbara * velocity_Barbara_y = (mass_Alfred + mass_Barbara) * velocity_y
Plugging in the known values:
(55 kg) * (2.17 m/s) = (83 kg + 55 kg) * velocity_y
Simplifying the equation:
(55 kg) * (2.17 m/s) = (138 kg) * velocity_y
velocity_y ≈ (55 kg * 2.17 m/s) / (138 kg) ≈ 0.86 m/s
Thus, the y-component of the velocity (vy) after the collision is approximately 0.86 m/s.
Step 4: Find the angle (θ) of the velocity.
The angle (θ) can be found using trigonometry:
tan(θ) = vy / vx
θ = arctan(vy / vx)
Plugging in the values:
θ ≈ arctan(0.86 m/s / 1.03 m/s)
θ ≈ arctan(0.837)
θ ≈ 40.1°
Therefore, the velocity (v) of the couple after the collision is approximately 1.03 m/s at an angle of 40.1° above the positive x-axis.