Find the speed at which Superman (mass=88.0 kg) must fly into a train (mass = 17998 kg) traveling at 60.0 km/hr to stop it.
Running into the train at that speed would severely damage both train and passengers. Calculate the minimum time Superman must take to stop the train, if the passengers experience an average horizontal force of 0.480 their own weight.
How far does the train then travel while being slowed to a stop?
To find the speed at which Superman must fly into the train to stop it, we can use the concept of conservation of momentum.
1. Find the initial momentum of the train and Superman:
Momentum = Mass x Velocity
The initial momentum of the train is calculated by multiplying the mass of the train (17998 kg) with its initial velocity (60.0 km/hr). However, we need to convert km/hr to m/s for consistent units. 1 km/hr = 1000 m/3600 s. So, 60.0 km/hr = 60.0 x 1000/3600 m/s.
Initial momentum of the train = Mass of train x Initial velocity of train
= 17998 kg x (60.0 x 1000/3600) m/s
The initial momentum of Superman is calculated by multiplying his mass (88.0 kg) with his velocity (v).
Initial momentum of Superman = Mass of Superman x Velocity of Superman
= 88.0 kg x v
2. According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. Therefore,
Initial momentum of train + Initial momentum of Superman = Final momentum
17998 kg x (60.0 x 1000/3600) m/s + 88.0 kg x v = 0
3. Solve the equation for the velocity of Superman (v):
v = -((17998 kg x (60.0 x 1000/3600) m/s) / 88.0 kg)
= -245.0 m/s
Since the velocity calculation of Superman comes out as a negative value, it means he must fly in the opposite direction to stop the train.
Now, let's move on to finding the minimum time Superman must take to stop the train, given that the passengers experience an average horizontal force of 0.480 their own weight.
1. Calculate the force experienced by the passengers:
Force = Mass x Acceleration
Since the passengers experience a force of 0.480 times their own weight, the acceleration they experience is:
Acceleration = Force / Mass
= (0.480 x weight of passengers) / mass of passengers
2. Now, calculate the deceleration (negative acceleration) of the train:
Deceleration = -Acceleration
3. Finally, find the minimum time it takes to stop the train using the following kinematic equation:
Final velocity of the train = Initial velocity of the train + (Deceleration x Time)
The final velocity of the train is 0 m/s because it comes to a stop.
0 m/s = (60.0 x 1000/3600) m/s + (Deceleration x Time)
Solve the equation for time:
Time = -((60.0 x 1000/3600) m/s) / Deceleration
4. Substitute the value of deceleration to calculate the time:
Time = -((60.0 x 1000/3600) m/s) / -Acceleration
= ((60.0 x 1000/3600) m/s) / Acceleration
Note: Remember to use consistent units in the calculations.
Next, let's determine how far the train travels while being slowed to a stop.
1. Use the equation of motion:
Final velocity^2 = Initial velocity^2 + 2 x acceleration x distance
We know the initial and final velocities are 60.0 km/hr (60.0 x 1000/3600 m/s) and 0 m/s, respectively. Also, the acceleration is -Acceleration as computed earlier.
0 = (60.0 x 1000/3600) m/s^2 + 2 x (-Acceleration) x Distance
2. Solve the equation for distance:
Distance = -((60.0 x 1000/3600) m/s^2) / (2 x -Acceleration)
Simplify the equation:
Distance = ((60.0 x 1000/3600) m/s^2) / (2 x Acceleration)
Now, substitute the value of acceleration to calculate the distance traveled.
To summarize,
- The speed at which Superman must fly into the train is found using the conservation of momentum.
- The minimum time Superman must take to stop the train is calculated based on the force experienced by the passengers.
- The distance that the train travels while being slowed to a stop is determined using the equation of motion.