A train on a mountain railway is carrying 200 people of average mass 70kg up a slope at an angle of 30° to the horizontal and at a speed of 6.0 m s–1. The train itself has a mass of 80 000 kg. The percentage of the power from the engine which is used to raise the passengers and the train is 40%.
What is the power of the engine?
mass of passengers=200*70=14000...total mass of train +passengers=80000+14000... their weight =94000*9.81=922140..now force = mgsin30= 922140sin30=461070.. power =force*velocity= 461070*6=2766420..this is power output.. now we know that power output divide by power input *100%=40%..solve it to get the answer 6.9MW...
To find the power of the engine, we need to calculate the total power used to raise the passengers and the train and then find the percentage of that power.
Step 1: Calculate the total mass being raised
The mass of the train is given as 80,000 kg, and there are 200 passengers with an average mass of 70 kg. Therefore, the total mass being raised is:
Total mass = mass of train + mass of passengers
Total mass = 80,000 kg + (200 passengers * 70 kg/passenger)
Total mass = 80,000 kg + 14,000 kg
Total mass = 94,000 kg
Step 2: Calculate the gravitational force acting on the train and passengers
The gravitational force acting on the train and passengers can be found using the formula:
Force = mass * g
where g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
Force = Total mass * g
Force = 94,000 kg * 9.8 m/s^2
Force = 921,200 N
Step 3: Calculate the work done to raise the train and passengers
The work done is equal to the force applied multiplied by the distance traveled. In this case, the distance is not given explicitly. However, the speed and angle of the slope can be used to calculate the vertical distance traveled.
Distance = speed * time
The time can be found using the angle of the slope and the speed.
Time = distance / speed = (vertical distance) / (speed * cos(angle))
Since the slope is at an angle of 30° to the horizontal, the vertical distance traveled can be calculated using:
Vertical distance = distance * sin(angle)
Distance = speed * time
Vertical distance = distance * sin(angle)
Vertical distance = (speed * time) * sin(angle)
Vertical distance = (6.0 m/s * time) * sin(30°)
Step 4: Calculate the time taken to travel the vertical distance
The time can be calculated using the horizontal distance traveled, which can be found using trigonometry.
Horizontal distance = distance * cos(angle)
Distance = speed * time
Horizontal distance = speed * time * cos(angle)
(speed * time) * cos(angle) = horizontal distance
time = (horizontal distance) / (speed * cos(angle))
time = (0) / (6.0 m/s * cos(30°))
In this case, the horizontal distance is 0 because the train is moving vertically uphill.
Step 5: Calculate the vertical distance traveled
Using the known values, we can calculate the vertical distance traveled:
Vertical distance = (6.0 m/s * time) * sin(30°)
Step 6: Calculate the work done
The work done is equal to the force applied multiplied by the distance traveled.
Work done = force * distance
Work done = 921,200 N * vertical distance
Step 7: Calculate the power used to raise the train and passengers
The power used is equal to the work done divided by the time taken.
Power used = work done / time
Step 8: Calculate the power of the engine
The power of the engine is given as 40% of the total power used to raise the train and passengers.
Power of engine = power used / 0.4
Now that we have the steps, let's calculate each value.
Step 1: Calculate the total mass being raised
Total mass = 94,000 kg
Step 2: Calculate the gravitational force acting on the train and passengers
Force = 921,200 N
Step 3: Calculate the work done to raise the train and passengers
Vertical distance = (6.0 m/s * time) * sin(30°)
Step 4: Calculate the time taken to travel the vertical distance
time = (0) / (6.0 m/s * cos(30°))
Step 5: Calculate the vertical distance traveled
Vertical distance = (6.0 m/s * time) * sin(30°)
Step 6: Calculate the work done
Work done = 921,200 N * vertical distance
Step 7: Calculate the power used to raise the train and passengers
Power used = work done / time
Step 8: Calculate the power of the engine
Power of engine = power used / 0.4
To find the power of the engine, we need to determine the work done in raising the passengers and the train up the slope.
The work done is given by the formula:
Work = Force × Distance × cos(θ),
where:
- Force is the component of force parallel to the direction of motion,
- Distance is the displacement of the object,
- θ is the angle between the force and displacement vectors.
In this case, the force is the combined weight of the passengers and the train, and the distance is the displacement along the inclined slope. The angle θ is given as 30°.
First, let's calculate the weight of the passengers and the train. The weight can be found using the formula:
Weight = mass × acceleration due to gravity,
where the acceleration due to gravity is approximately 9.8 m/s².
The weight of the passengers is 200 × 70 kg × 9.8 m/s², and the weight of the train is 80,000 kg × 9.8 m/s².
Next, we need to find the component of the weight force parallel to the direction of motion. This can be calculated using the formula:
Force parallel = weight × sin(θ),
where θ is the angle between the slope and the horizontal direction.
Now, we can calculate the work done:
Work = Force parallel × Distance × cos(θ).
The work done represents the energy transferred over a certain time interval, which is the definition of power.
Finally, the power of the engine can be found by dividing the work done by the time interval it took to complete:
Power = Work / Time.
Now that we have explained the process, let's calculate the answer.
Power=Mg(speed)*sin(θ)/0.40 watts
M=total mass (passengers, train, etc.)