A 2.0 kg mass is moving on a frictionless airtrack. It collides into a motionless 1.5 kg mass. What is the combined speed of the two masses if they stick together on impact?
To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
The momentum (p) of an object is given by the product of its mass (m) and velocity (v), i.e., p = m * v.
Let's assume the initial velocity of the 2.0 kg mass (m1) is v1 and the initial velocity of the 1.5 kg mass (m2) is v2. Since the 1.5 kg mass is motionless, its initial velocity is 0 (v2 = 0).
The total momentum before the collision is the sum of the momentum of the two masses:
p_initial = p1_initial + p2_initial
= m1 * v1 + m2 * v2
= 2.0 kg * v1 + 1.5 kg * 0
= 2.0 kg * v1
After the collision, the two masses stick together, so they move with a common velocity, let's call it V.
The total momentum after the collision is:
p_final = p_combined
= (m1 + m2) * V
= (2.0 kg + 1.5 kg) * V
= 3.5 kg * V
Since the principle of conservation of momentum states that the initial momentum is equal to the final momentum:
p_initial = p_final
Plugging in the values, we have:
2.0 kg * v1 = 3.5 kg * V
To find the combined speed (V) of the two masses, we can rearrange the equation:
V = (2.0 kg * v1) / 3.5 kg
Therefore, the combined speed of the two masses after the collision is given by (2.0 kg * v1) / 3.5 kg.
To find the combined speed of the two masses after the collision, we can use the law of conservation of momentum. According to this law, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by its mass multiplied by its velocity. Suppose the initial velocity of the 2.0 kg mass is v1, and the initial velocity of the 1.5 kg mass is 0 (since it is motionless). After the collision, the masses stick together, meaning they move as one body. Let's call their combined velocity v2.
Before the collision, the total momentum is given by:
Initial momentum = (mass1 * velocity1) + (mass2 * velocity2)
After the collision, the total momentum is given by:
Final momentum = (combined mass * combined velocity)
Since momentum is conserved, we can write the equation:
(mass1 * velocity1) + (mass2 * velocity2) = (combined mass * combined velocity)
Plugging in the given values:
(2.0 kg * v1) + (1.5 kg * 0) = (2.0 kg + 1.5 kg) * v2
Simplifying the equation:
2.0 kg * v1 = 3.5 kg * v2
(v1) = (3.5 kg * v2) / (2.0 kg)
Since the masses stick together, their final velocity is the same, so we can equate v1 to v2:
v1 = v2
Substituting v2:
v1 = (3.5 kg * v1) / (2.0 kg)
Now we can solve for v1:
v1 * (2.0 kg) = (3.5 kg) * v1
2.0 kg * v1 = 3.5 kg * v1
Dividing both sides by v1 (assuming v1 ≠ 0):
v1 = 3.5 kg / 2.0 kg
v1 = 1.75 m/s
Therefore, the combined speed of the two masses after sticking together is 1.75 m/s.