Find the speed at which Superman (mass=89.0 kg) must fly into a train (mass = 19564 kg) traveling at 65.0 km/hr to stop it.
The system (train + Superman) must end up with zero momentum. With no external forces, Superman's initial momentum must be equal and oppositely directed to the train's. (This is not good for the passengers, as they stop too quickly.)
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 0.520 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 principle of conservation of momentum. The total momentum before the collision must be equal to the total momentum after the collision. Since Superman and the train are initially moving in opposite directions, the momentum of the train is negative.
Momentum = mass × velocity
The momentum of Superman before the collision is given by:
Superman's momentum before = mass of Superman × velocity of Superman before
And the momentum of the train before the collision is given by:
Train's momentum before = mass of the train × velocity of the train before
Since the system must end up with zero momentum:
Superman's momentum before + Train's momentum before = 0
Substituting the given values:
mass of Superman × velocity of Superman before + mass of the train × velocity of the train before = 0
Solving for the velocity of Superman before:
velocity of Superman before = - (mass of the train × velocity of the train before) / mass of Superman
Now we can plug in the values:
mass of Superman = 89.0 kg
mass of the train = 19564 kg
velocity of the train before = 65.0 km/hr
First, let's convert the velocity of the train to m/s:
velocity of the train before = 65.0 km/hr × (1000 m / 1 km) × (1 hr / 3600 s)
= 18.0556 m/s
Now we can substitute the values into the equation and calculate the velocity of Superman before:
velocity of Superman before = - (19564 kg × 18.0556 m/s) / 89.0 kg
Calculating the velocity of Superman before:
velocity of Superman before = - 3971.8732 m/s
Since the velocity is negative, it means Superman must fly into the train in the opposite direction of its initial velocity.
To find the minimum time Superman must take to stop the train, we can use the equation for average force:
Average force = Change in momentum / Time
The change in momentum can be calculated as:
Change in momentum = momentum after - momentum before
The momentum after the collision is zero because the system comes to a stop. Thus, the change in momentum is equal to the momentum before the collision.
Change in momentum = momentum before
Now we can use the equation for average force to find the time needed to stop the train:
Average force = 0.520 × Weight of passengers
Weight of passengers = mass of the train × gravitational acceleration
Gravitational acceleration = 9.8 m/s^2
Substituting the values:
Average force = 0.520 × (mass of the train × gravitational acceleration)
Average force = (0.520 × 19564 kg × 9.8 m/s^2) / Time
Since the average force is equal to the change in momentum, we can set these two equations equal to each other:
(0.520 × 19564 kg × 9.8 m/s^2) / Time = momentum before
Solving for Time:
Time = (0.520 × 19564 kg × 9.8 m/s^2) / momentum before
Plugging in the values:
momentum before = mass of the train × velocity of the train before
Time = (0.520 × 19564 kg × 9.8 m/s^2) / (mass of the train × velocity of the train before)
Calculating the time needed to stop the train:
Time = (0.520 × 19564 kg × 9.8 m/s^2) / (19564 kg × 18.0556 m/s)
Now to find the distance the train travels while being slowed to a stop, we can use the equation for distance:
Distance = (1/2) × acceleration × time^2
Acceleration = -velocity of the train before / Time (negative because the train is slowing down)
Substituting the values:
Distance = (1/2) × (-velocity of the train before / Time) × time^2
Calculating the distance:
Distance = (1/2) × (-18.0556 m/s / Time) × (Time^2)
Substituting the calculated value for Time:
Distance = (1/2) × (-18.0556 m/s / [(0.520 × 19564 kg × 9.8 m/s^2) / (19564 kg × 18.0556 m/s)]) × ([(0.520 × 19564 kg × 9.8 m/s^2) / (19564 kg × 18.0556 m/s)]^2)