One object is at rest and another is moving. The two collide in a one-dimensional,
completely inelastic collision. In other words, they stick together and move off with
shared velocity. The speed of the object that is moving initially is 25 m/s. The masses of
the two objects are 3.0 and 8.0 kg. Determine the final speed of the two-object system
after the collision for the case when the large-mass object is the one moving initially and
the case when the small-mass object is the one moving initially.
Case #1:
Momentum = m1V1-m2V2 =8*25-3*0 = 200
Momentum = (m1+m2)V = 200
(8+3)*V = 200
V = 18.18 m/s
Case #2:
Momentum. = 3*25-8*0 = 75
(m1+m2)*V = 75
(8+3)*V = 75
V = 6.82 m/s
To determine the final velocity of the two-object system after the collision, we can use the principle of conservation of momentum.
The momentum before the collision is given by:
initial momentum = mass₁ * velocity₁ + mass₂ * velocity₂
Case 1: Large-mass object is moving initially
Mass of object 1 (large-mass object) = 8.0 kg
Mass of object 2 (small-mass object) = 3.0 kg
Velocity of object 1 (large-mass object) = 25 m/s
Velocity of object 2 (small-mass object) = 0 m/s (at rest)
Initial momentum = (8.0 kg) * (25 m/s) + (3.0 kg) * (0 m/s) = 200 kg·m/s
After the collision, the two objects stick together and move off with a shared velocity, so the final velocity of the two-object system can be calculated as:
final velocity = (mass₁ * velocity₁ + mass₂ * velocity₂) / (mass₁ + mass₂)
final velocity = (8.0 kg * 25 m/s + 3.0 kg * 0 m/s) / (8.0 kg + 3.0 kg)
final velocity = (200 kg·m/s) / (11 kg)
final velocity ≈ 18.18 m/s
Therefore, in Case 1, when the large-mass object is initially moving, the final speed of the two-object system is approximately 18.18 m/s.
Case 2: Small-mass object is moving initially
Mass of object 1 (large-mass object) = 8.0 kg
Mass of object 2 (small-mass object) = 3.0 kg
Velocity of object 1 (large-mass object) = 0 m/s (at rest)
Velocity of object 2 (small-mass object) = 25 m/s
Initial momentum = (8.0 kg) * (0 m/s) + (3.0 kg) * (25 m/s) = 75 kg·m/s
After the collision, the two objects stick together and move off with a shared velocity, so the final velocity of the two-object system can be calculated as:
final velocity = (mass₁ * velocity₁ + mass₂ * velocity₂) / (mass₁ + mass₂)
final velocity = (8.0 kg * 0 m/s + 3.0 kg * 25 m/s) / (8.0 kg + 3.0 kg)
final velocity = (75 kg·m/s) / (11 kg)
final velocity ≈ 6.82 m/s
Therefore, in Case 2, when the small-mass object is initially moving, the final speed of the two-object system is approximately 6.82 m/s.
To determine the final speed of the two-object system after the collision, we can use the law 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, assuming there are no external forces acting on the system.
Momentum (p) is defined as the product of an object's mass and its velocity: p = m * v, where m is the mass and v is the velocity.
Let's calculate the momentum before the collision in both cases:
Case 1: Large-mass object is moving initially
Mass of the large-mass object (m1) = 8.0 kg
Initial velocity of the large-mass object (v1) = 25 m/s
Momentum of the large-mass object (p1) = m1 * v1 = 8.0 kg * 25 m/s = 200 kg m/s
Mass of the small-mass object (m2) = 3.0 kg
Initial velocity of the small-mass object (v2) = 0 m/s (at rest)
Momentum of the small-mass object (p2) = m2 * v2 = 3.0 kg * 0 m/s = 0 kg m/s
Total momentum before the collision (p_initial) = p1 + p2 = 200 kg m/s + 0 kg m/s = 200 kg m/s
Case 2: Small-mass object is moving initially
Mass of the small-mass object (m1) = 3.0 kg
Initial velocity of the small-mass object (v1) = 25 m/s
Momentum of the small-mass object (p1) = m1 * v1 = 3.0 kg * 25 m/s = 75 kg m/s
Mass of the large-mass object (m2) = 8.0 kg
Initial velocity of the large-mass object (v2) = 0 m/s (at rest)
Momentum of the large-mass object (p2) = m2 * v2 = 8.0 kg * 0 m/s = 0 kg m/s
Total momentum before the collision (p_initial) = p1 + p2 = 75 kg m/s + 0 kg m/s = 75 kg m/s
Now, let's find the final velocity of the two-object system after the collision using the law of conservation of momentum:
Case 1: Large-mass object is moving initially
Final velocity of the two-object system (v_final) = p_initial / (m1 + m2)
v_final = 200 kg m/s / (8.0 kg + 3.0 kg) = 200 kg m/s / 11.0 kg
v_final ≈ 18.18 m/s
Case 2: Small-mass object is moving initially
Final velocity of the two-object system (v_final) = p_initial / (m1 + m2)
v_final = 75 kg m/s / (3.0 kg + 8.0 kg) = 75 kg m/s / 11.0 kg
v_final ≈ 6.82 m/s
Therefore, the final speed of the two-object system after the collision is approximately 18.18 m/s when the large-mass object is the one moving initially, and approximately 6.82 m/s when the small-mass object is the one moving initially.