Boxcar A, with mass of 1500 kg, is travelling at 25 m/s to the east. Boxcar B has a mass of 2000 kg, and is initially at rest. The box cars collide in elastically and move together after they get stuck. What is their combined velocity?
MV+MV=(M+M)V
(1,500Kg)(25m/s)+(2,000Kg)(0)= (2000Kg+1500Kg)V
37,500=3,500V
37,500/3,500=3,500/3,500V
V=1.07m/s
10.7m/s
To find the combined velocity of the two boxcars after the collision, 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.
The momentum of an object is given by the product of its mass and velocity. Therefore, we can calculate the initial momentum of Boxcar A and Boxcar B.
The initial momentum of Boxcar A (p₁) is equal to the product of its mass (m₁) and velocity (v₁):
p₁ = m₁ * v₁
The initial momentum of Boxcar B (p₂) is equal to the product of its mass (m₂) and velocity (v₂):
p₂ = m₂ * v₂
Since Boxcar B is initially at rest (v₂ = 0), its initial momentum (p₂) is zero.
According to the principle of conservation of momentum, the total momentum before the collision (p_total) is equal to the total momentum after the collision (p_total'). Therefore, we have:
p_total = p₁ + p₂ = m₁ * v₁ + 0
After the collision, the two boxcars become stuck together, so they move as a single object. Let's denote the combined mass of the two boxcars as (m_total) and the combined velocity as (v_total).
The total momentum after the collision (p_total') is equal to the product of the combined mass (m_total) and the combined velocity (v_total):
p_total' = m_total * v_total
Since the collision is elastic, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
The kinetic energy of an object is given by the formula: E_kinetic = (1/2) * m * v^2
The total initial kinetic energy (E_total) is the sum of the kinetic energies of Boxcar A and Boxcar B:
E_total = (1/2) * m₁ * v₁^2 + (1/2) * m₂ * v₂^2
The total final kinetic energy (E_total') is the kinetic energy of the combined boxcars:
E_total' = (1/2) * m_total * v_total^2
Since the collision is elastic, E_total = E_total', so we have:
(1/2) * m₁ * v₁^2 + (1/2) * m₂ * v₂^2 = (1/2) * m_total * v_total^2
Now, let's substitute the given values into the equations and solve for the combined velocity (v_total).
Given:
m₁ = 1500 kg
v₁ = 25 m/s
m₂ = 2000 kg
First, calculate the initial momentum of Boxcar A:
p₁ = m₁ * v₁ = 1500 kg * 25 m/s = 37500 kg·m/s
Since Boxcar B is initially at rest, its initial momentum is zero:
p₂ = 0
The total momentum before the collision is equal to the momentum of Boxcar A:
p_total = p₁ = 37500 kg·m/s
Now, calculate the combined mass of the two boxcars:
m_total = m₁ + m₂ = 1500 kg + 2000 kg = 3500 kg
Since the total momentum after the collision is equal to the total momentum before the collision, we have:
p_total' = p_total = 37500 kg·m/s
Now, let's rearrange the equation for total momentum after the collision to solve for the combined velocity (v_total):
p_total' = m_total * v_total
37500 kg·m/s = 3500 kg * v_total
Divide both sides of the equation by 3500 kg:
v_total = 37500 kg·m/s / 3500 kg
v_total ≈ 10.714 m/s
Therefore, the combined velocity of the two boxcars after the collision is approximately 10.714 m/s.