a box (mass = 12 kg ) is going south at 12 m/s. A second box (mass = 18 kg ) is going northwest at 15 m/s. they collide and stick together. what is their common velocity (magnitude and direction ) after the collision
mv1 = <0,-144>
mv2 = <-270/√2, 270/√2>
add the vectors and then change back to direction and magnitude.
To find the common velocity of the two boxes after the collision, we'll need to apply the principles of conservation of momentum. The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision.
Before the collision, we need to calculate the momentum of each box separately. Momentum (p) is calculated using the formula p = m * v, where m is the mass, and v is the velocity.
For the first box:
Mass (m1) = 12 kg
Velocity (v1) = 12 m/s
Momentum of the first box (p1) = m1 * v1 = 12 kg * 12 m/s = 144 kg·m/s (southward direction)
For the second box:
Mass (m2) = 18 kg
Velocity (v2) = 15 m/s
The second box is moving in the northwest direction, which can be considered as a combination of motion in the north and west directions. To calculate its momentum, we need to find the north and west components separately.
The north component of the velocity (v2n) can be calculated as v2n = v2 * cos(45°), where 45° is the angle of motion with respect to the north. Since cos(45°) = sin(45°) = 1/√2, we have:
v2n = 15 m/s * (1/√2) = (15/√2) m/s ≈ 10.61 m/s (northward direction)
The west component of the velocity (v2w) can be calculated as v2w = v2 * sin(45°), which is equal to:
v2w = 15 m/s * (1/√2) = (15/√2) m/s ≈ 10.61 m/s (westward direction)
Now that we have the north and west components of the velocity, we can calculate the momentum of the second box:
Momentum of the second box (p2) = m2 * (v2n + v2w) = 18 kg * (10.61 m/s + 10.61 m/s) = 381.78 kg·m/s (northwest direction)
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
Total momentum before the collision = p1 + p2
Total momentum after the collision = (m1 + m2) * v
Since the two boxes stick together, their masses are combined. The total mass (m_total) after the collision is m1 + m2 = 12 kg + 18 kg = 30 kg.
Therefore, the equation becomes:
Total momentum before the collision = Total momentum after the collision
p1 + p2 = m_total * v
Substituting the known values:
144 kg·m/s + 381.78 kg·m/s = 30 kg * v
525.78 kg·m/s = 30 kg * v
To find the common velocity (v), divide both sides of the equation by the mass:
v = (p1 + p2) / m_total
v = (144 kg·m/s + 381.78 kg·m/s) / 30 kg
v ≈ 17.53 m/s
Therefore, the common velocity of the two boxes after the collision is approximately 17.53 m/s. To determine the direction, we examine the vectors involved:
- The momentum of the first box before the collision is in the southward direction.
- The momentum of the second box before the collision is in the northwest direction.
Since the two boxes stick together, their resultant momentum will be a combination of these two momenta. Thus, the common velocity after the collision will be in a direction between south and northwest.