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
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
To find the final velocity at which the two cars move forward after the collision, we can use the principle of conservation of momentum.
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is calculated by multiplying its mass by its velocity.
Let's calculate the momentum of each car before the collision:
Car 1:
Mass1 = 1250 kg
Velocity1 = 2.24 m/s
Momentum1 = Mass1 * Velocity1
Car 2:
Mass2 = 1300 kg
Velocity2 = 0 m/s (since the car is initially at rest)
Momentum2 = Mass2 * Velocity2
Total momentum before the collision = Momentum1 + Momentum2
After the collision, the two cars stick together and move forward with a final velocity, let's call it Vf.
Total momentum after the collision = (Mass1 + Mass2) * Vf
According to the principle of conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision:
Momentum1 + Momentum2 = (Mass1 + Mass2) * Vf
Substituting the values we have:
(Mass1 * Velocity1) + (Mass2 * Velocity2) = (Mass1 + Mass2) * Vf
(1250 kg * 2.24 m/s) + (1300 kg * 0 m/s) = (1250 kg + 1300 kg) * Vf
2800 kg·m/s = 2550 kg * Vf
Vf = 2800 kg·m/s / 2550 kg
Vf ≈ 1.10 m/s
Therefore, after the collision, the two cars will move forward with a velocity of approximately 1.10 m/s.
To find the velocity at which the two cars move forward after the collision, 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 of an object can be calculated by multiplying its mass by its velocity. So, the momentum of the first car (car 1) before the collision is given by:
Momentum of car 1 before collision = mass of car 1 * velocity of car 1
Similarly, the momentum of the second car (car 2) before the collision is given by:
Momentum of car 2 before collision = mass of car 2 * velocity of car 2
Since the two cars stick together and move forward after the collision, we can treat them as a single combined object. The total mass of the system after the collision is the sum of the masses of car 1 and car 2. The velocity of the combined object can be represented by "v".
So, the total momentum after the collision is given by:
Total momentum after collision = (mass of car 1 + mass of car 2) * velocity
As per the principle of conservation of momentum:
Momentum of car 1 before collision + Momentum of car 2 before collision = Total momentum after collision
(mass of car 1 * velocity of car 1) + (mass of car 2 * velocity of car 2) = (mass of car 1 + mass of car 2) * velocity
Now, let's substitute the given values into the equation:
(1250 kg * 2.24 m/s) + (1300 kg * 0 m/s) = (1250 kg + 1300 kg) * velocity
2800 + 0 = 2550 * velocity
2800 = 2550 * velocity
To find the velocity at which the two cars move forward after the collision, we need to solve for "velocity" by dividing both sides of the equation by 2550:
velocity = 2800 / 2550
velocity ≈ 1.10 m/s
Therefore, the two cars will move forward at approximately 1.10 m/s after the collision.