To solve this problem, we can use the law of conservation of momentum, which states that the total momentum before a collision is equal to the total momentum after the collision.
Let's consider the velocity of the first car before the collision as v₁ (2.24 m/s) and the velocity of the second car before the collision as v₂ (unknown). After the collision, the two cars stick together and move forward with a final velocity v.
According to the law of conservation of momentum:
(m₁ * v₁) + (m₂ * v₂) = (m₁ + m₂) * v
Where:
m₁ = mass of the first car (1250 kg)
v₁ = velocity of the first car before the collision (2.24 m/s)
m₂ = mass of the second car (1300 kg)
v₂ = velocity of the second car before the collision (unknown)
m₁ + m₂ = total mass after the collision (m₁ + m₂ = 1250 kg + 1300 kg = 2550 kg)
v = final velocity of both cars after the collision (unknown)
Substituting the given values into the equation:
(1250 kg * 2.24 m/s) + (1300 kg * v₂) = (2550 kg) * v
2800 kg·m/s + (1300 kg * v₂) = 2550 kg * v
Next, we need to use the principle of conservation of kinetic energy to determine the velocity v₂ of the second car before the collision. Kinetic energy is also conserved in an elastic collision.
According to the conservation of kinetic energy,
(1/2 * m₁ * v₁²) + (1/2 * m₂ * v₂²) = (1/2 * (m₁ + m₂) * v²)
Substituting the given values into the equation:
(1/2 * 1250 kg * (2.24 m/s)²) + (1/2 * 1300 kg * v₂²) = (1/2 * 2550 kg * v²)
(1/2 * 1250 kg * 5.0176 m²/s²) + (1/2 * 1300 kg * v₂²) = (1/2 * 2550 kg * v²)
3134.5 kg·m²/s² + (1/2 * 1300 kg * v₂²) = 1275 kg·v²
Now we can solve these two equations simultaneously to find the values of v and v₂.
2800 kg·m/s + (1300 kg * v₂) = 2550 kg * v -----(1)
3134.5 kg·m²/s² + (1/2 * 1300 kg * v₂²) = 1275 kg·v² -----(2)
With some algebraic manipulation, we can rearrange equation (1) to isolate v:
v = (2800 kg·m/s + (1300 kg * v₂)) / 2550 kg
Now substitute this value of v into equation (2) to solve for v₂:
3134.5 kg·m²/s² + (1/2 * 1300 kg * v₂²) = 1275 kg·(2800 kg·m/s + (1300 kg * v₂))^2 / 2550 kg²
Simplifying this equation and solving for v₂ will give us the velocity of the second car before the collision. Once we have v₂, we can calculate the final velocity, v, by substituting the values of v₁, v₂, m₁, and m₂ into equation (1):
v = (2800 kg·m/s + (1300 kg * v₂)) / 2550 kg
A car with a mass of 1250 kg at 2.24 m/s and bumps into a car With a mass of 1300 kg. After the collision the two cars stick together and move forward. How fast will they both move forward
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