Two bodies of masses 8 kg and 4 kg move along the x-axis in opposite directions with velocities of 11 m/s-positive x-direction and 7 m/s-negative x-direction, respectively. They collide and stick together. Find their center of mass. Show the solution and answer.
To find the center of mass of the system, we need to consider the individual masses and their positions.
Let's denote the mass of the first body as m1 (8 kg) and its velocity as v1 (11 m/s in the positive x-direction). The mass of the second body is m2 (4 kg) and its velocity is v2 (-7 m/s in the negative x-direction).
First, we need to calculate the momentum of each body. The momentum (p) of a body is given by the product of its mass and velocity: p = m * v.
For the first body:
p1 = m1 * v1 = 8 kg * 11 m/s = 88 kg·m/s
For the second body:
p2 = m2 * v2 = 4 kg * (-7 m/s) = -28 kg·m/s (Note: we include the negative sign as the velocity is in the opposite direction to the positive x-axis.)
During the collision, the two bodies stick together, so their combined mass will be m1 + m2 = 8 kg + 4 kg = 12 kg.
To find the center of mass, we need to consider the weighted average of the positions of the two bodies based on their masses.
Let's assume the initial position of the first body was x1, and the initial position of the second body was x2.
The center of mass (x_cm) is given by the formula:
x_cm = (m1 * x1 + m2 * x2) / (m1 + m2)
In this case, since the two bodies collide and stick together, they will move together as a single object. The final velocity (v_f) of this object after sticking together is given by the conservation of momentum:
p_f = p1 + p2 = (m1 + m2) * v_f
Let's denote this final velocity as v_f, and the final position as x_f.
So, we have the equation:
(m1 + m2) * v_f = m1 * v1 + m2 * v2
Substituting the values:
12 kg * v_f = 8 kg * 11 m/s + 4 kg * (-7 m/s)
Simplifying:
12 kg * v_f = 88 kg·m/s - 28 kg·m/s
12 kg * v_f = 60 kg·m/s
v_f = 60 kg·m/s / 12 kg
v_f = 5 m/s
Since the final velocity (v_f) is the velocity of the combined object, we can use it to find the final position (x_f) using the formula:
x_f = x_cm
Now, let's substitute the given initial positions and the final velocity into the formula for the center of mass:
x_cm = (m1 * x1 + m2 * x2) / (m1 + m2)
For the first body, x1 = 0 m (assuming it starts at the origin).
For the second body, x2 = -0 m since it moves in the negative x-direction.
Substituting the values:
x_cm = (8 kg * 0 m + 4 kg * (-0 m)) / (8 kg + 4 kg)
x_cm = 0 m / 12 kg
x_cm = 0 m
Therefore, the center of mass of the system is located at x = 0 m.
In summary, the center of mass of the system of two bodies is located at x = 0 m.