In a railroad switchyard, a 48ton freight car is sent at 6.8mi/hr toward a 28ton car that is moving in the same direction at 3.8mi/hr .What is the speed of the pair after they couple together?What fraction of the initial kinetic energy was lost in the collision?
This problem is hella stupid man. I can't even solve it!
To find the speed of the pair after they couple together, we can use the principle of conservation of momentum.
The momentum of an object is given by the product of its mass and velocity. When the two cars couple together, the total momentum before and after the collision should be the same.
Let's calculate the initial and final momenta separately:
Initial momentum:
Momentum of the first car = mass of the first car * velocity of the first car
= 48 tons * 6.8 mi/hr
Momentum of the second car = mass of the second car * velocity of the second car
= 28 tons * 3.8 mi/hr
Total initial momentum = initial momentum of first car + initial momentum of second car
To calculate the final speed of the couple, we divide the total initial momentum by the sum of the masses of the two cars.
Final momentum:
Momentum of the couple = Total mass of the two cars * final velocity of the couple
Now, let's calculate the values:
Step 1: Convert the masses and velocities to the same units:
1 ton = 2000 pounds
1 mile = 5280 feet
1 hour = 3600 seconds
Mass of the first car = 48 tons = 48 * 2000 pounds
Mass of the second car = 28 tons = 28 * 2000 pounds
Step 2: Convert the velocities from miles per hour to feet per second:
Velocity of the first car = 6.8 mi/hr = 6.8 * 5280 feet / 3600 seconds
Velocity of the second car = 3.8 mi/hr = 3.8 * 5280 feet / 3600 seconds
Step 3: Calculate the momenta:
Initial momentum of the first car = Mass of the first car * Velocity of the first car
Initial momentum of the second car = Mass of the second car * Velocity of the second car
Total initial momentum = Initial momentum of the first car + Initial momentum of the second car
Step 4: Calculate the final velocity of the couple:
Final velocity of the couple = Total initial momentum / (Mass of the first car + Mass of the second car)
Step 5: Calculate the fraction of kinetic energy lost:
Fraction of kinetic energy lost = (Initial kinetic energy - Final kinetic energy) / Initial kinetic energy
Let's plug in the values and calculate:
Mass of the first car = 48 tons = 48 * 2000 pounds = 96,000 pounds
Mass of the second car = 28 tons = 28 * 2000 pounds = 56,000 pounds
Velocity of the first car = 6.8 mi/hr = 6.8 * 5280 feet / 3600 seconds = 9.9266 ft/s (rounded to 4 decimal places)
Velocity of the second car = 3.8 mi/hr = 3.8 * 5280 feet / 3600 seconds = 5.5782 ft/s (rounded to 4 decimal places)
Initial momentum of the first car = 96,000 pounds * 9.9266 ft/s = 951,993.6 pound-ft/s (rounded to 1 decimal place)
Initial momentum of the second car = 56,000 pounds * 5.5782 ft/s = 312,189.2 pound-ft/s (rounded to 1 decimal place)
Total initial momentum = 951,993.6 pound-ft/s + 312,189.2 pound-ft/s = 1,264,182.8 pound-ft/s (rounded to 1 decimal place)
Final velocity of the couple = 1,264,182.8 pound-ft/s / (96,000 pounds + 56,000 pounds) = 7.9196 ft/s (rounded to 4 decimal places)
Now, let's calculate the fraction of kinetic energy lost:
Initial kinetic energy = 0.5 * Mass of the first car * (Velocity of the first car)^2 + 0.5 * Mass of the second car * (Velocity of the second car)^2
Final kinetic energy = 0.5 * (Mass of the first car + Mass of the second car) * (Final velocity of the couple)^2
Fraction of kinetic energy lost = (Initial kinetic energy - Final kinetic energy) / Initial kinetic energy
Let's calculate the values:
Initial kinetic energy = 0.5 * 96,000 pounds * (9.9266 ft/s)^2 + 0.5 * 56,000 pounds * (5.5782 ft/s)^2
Final kinetic energy = 0.5 * (96,000 pounds + 56,000 pounds) * (7.9196 ft/s)^2
Fraction of kinetic energy lost = (Initial kinetic energy - Final kinetic energy) / Initial kinetic energy
By calculating the above values, you will be able to find the speed of the pair after they couple together and the fraction of the initial kinetic energy lost in the collision.
To find the speed of the pair after they couple together, we need to apply the law of conservation of momentum, which states that 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. So, let's calculate the momentum of each car first.
Momentum of the 48-ton freight car:
Mass = 48 tons = 48000 kg (since 1 ton = 1000 kg)
Velocity = 6.8 mi/hr
Momentum = Mass × Velocity
Momentum of the freight car = 48000 kg × (6.8 mi/hr × 0.44704 m/s/mi)
Momentum of the freight car = 151009.92 kg·m/s
Momentum of the 28-ton car:
Mass = 28 tons = 28000 kg
Velocity = 3.8 mi/hr
Momentum = Mass × Velocity
Momentum of the 28-ton car = 28000 kg × (3.8 mi/hr × 0.44704 m/s/mi)
Momentum of the 28-ton car = 44704.64 kg·m/s
Now, to find the total momentum before the collision, we add the individual momentum of each car:
Total momentum before the collision = Momentum of freight car + Momentum of 28-ton car
Total momentum before the collision = 151009.92 kg·m/s + 44704.64 kg·m/s
Total momentum before the collision = 195714.56 kg·m/s
Since the two cars couple together after the collision, their masses combine, and therefore:
Total mass after the collision = Mass of freight car + Mass of 28-ton car
Total mass after the collision = 48000 kg + 28000 kg
Total mass after the collision = 76000 kg
Now, we can calculate the speed of the pair after they couple together using the formula for momentum:
Total momentum after the collision = Total mass after the collision × Speed after the collision
Since the two cars will have the same speed after they couple together, we can denote the speed as 'v':
195714.56 kg·m/s = 76000 kg × v
Solving for 'v', we find:
v = 195714.56 kg·m/s / 76000 kg
v ≈ 2.572 m/s
Therefore, the speed of the pair after they couple together is approximately 2.572 m/s.
To find the fraction of the initial kinetic energy lost in the collision, we can use the concept of kinetic energy.
The initial kinetic energy is given by the sum of the kinetic energy of each car before the collision:
Initial kinetic energy = (1/2) × Mass of the freight car × (Velocity of the freight car)^2 + (1/2) × Mass of the 28-ton car × (Velocity of the 28-ton car)^2
Substituting the given values:
Initial kinetic energy = (1/2) × 48000 kg × (6.8 mi/hr × 0.44704 m/s/mi)^2 + (1/2) × 28000 kg × (3.8 mi/hr × 0.44704 m/s/mi)^2
Calculating the initial kinetic energy:
Initial kinetic energy ≈ 686340.00 kg·m²/s² + 478060.48 kg·m²/s²
Initial kinetic energy ≈ 1164400.48 kg·m²/s²
The final kinetic energy after the collision is given by:
Final kinetic energy = (1/2) × (Total mass after the collision) × (Speed after the collision)^2
Substituting the values:
Final kinetic energy = (1/2) × 76000 kg × (2.572 m/s)^2
Calculating the final kinetic energy:
Final kinetic energy ≈ 246716.59 kg·m²/s²
To find the fraction of the initial kinetic energy lost, we can divide the change in kinetic energy by the initial kinetic energy:
Fraction of initial kinetic energy lost = (Initial kinetic energy - Final kinetic energy) / Initial kinetic energy
Fraction of initial kinetic energy lost = (1164400.48 kg·m²/s² - 246716.59 kg·m²/s²) / 1164400.48 kg·m²/s²
Calculating the fraction:
Fraction of initial kinetic energy lost ≈ 0.788
Therefore, the fraction of the initial kinetic energy lost in the collision is approximately 0.788, or 78.8%.