A 5.0 kg object moving in the +x direction at 5.5 m/s collides head-on with a 3.1 kg object moving in the -x direction at 4.0 m/s. Find the final velocity of each mass for each of the following situations. (Take the positive direction to be +x.)
The bodies stick together.
m/s (5.0 kg mass)
m/s (3.1 kg mass)
To find the final velocity of each mass when they stick together, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
Let's first calculate the total momentum before the collision:
Total momentum before = (mass1 * velocity1) + (mass2 * velocity2)
= (5.0 kg * 5.5 m/s) + (3.1 kg * (-4.0 m/s))
= 27.5 kg*m/s + (-12.4 kg*m/s)
= 15.1 kg*m/s
Since the two masses stick together, their combined mass after the collision is the sum of the individual masses:
Combined mass = mass1 + mass2
= 5.0 kg + 3.1 kg
= 8.1 kg
Therefore, the final velocity of the two masses (which are now stuck together) can be calculated by dividing the total momentum by the combined mass:
Final velocity = Total momentum / Combined mass
= 15.1 kg*m/s / 8.1 kg
= 1.86 m/s (rounded to two decimal places)
So, the final velocity of the combined masses is 1.86 m/s. Since the two objects stick together, this final velocity is the same for both masses.
Therefore, the final velocity of the 5.0 kg mass and the 3.1 kg mass, when they stick together, is 1.86 m/s.
To find the final velocity of each mass when the bodies stick together after the collision, we need to apply the principle of conservation of momentum. The total momentum before the collision should equal the total momentum after the collision.
The momentum (p) of an object is calculated by multiplying its mass (m) with its velocity (v). Mathematically, it can be written as:
p = m * v
Let's calculate the total momentum before the collision:
Initial momentum of the 5.0 kg object (moving in +x direction) = 5.0 kg * 5.5 m/s = 27.5 kg*m/s (in the +x direction)
Initial momentum of the 3.1 kg object (moving in -x direction) = 3.1 kg * (-4.0 m/s) = -12.4 kg*m/s (in the -x direction)
The total momentum before the collision is the vector sum of these two momenta:
Total momentum before the collision = 27.5 kg*m/s - 12.4 kg*m/s = 15.1 kg*m/s (in the +x direction)
Now, because the bodies stick together after the collision, their final velocity will be the same. Let's assume this common final velocity is V.
The total momentum after the collision is the product of the combined mass and the common final velocity:
Total momentum after the collision = (5.0 kg + 3.1 kg) * V = 8.1 kg * V
According to the conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision:
15.1 kg*m/s = 8.1 kg * V
Now, we can solve this equation to find the common final velocity (V):
V = 15.1 kg*m/s / 8.1 kg
V ≈ 1.86 m/s
Therefore, the final velocity of both masses when they stick together is approximately 1.86 m/s in the +x direction.
So, the final velocities are:
1.86 m/s for the 5.0 kg mass
1.86 m/s for the 3.1 kg mass
The bodies stick together: non elastic, use conservation of momentum.
M1*V1+M2*V2=(M1+M2)Vf