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 principle of conservation of momentum. The total momentum before the collision will be equal to the total momentum after the collision.
Given:
Mass of Superman, m1 = 88.0 kg
Mass of train, m2 = 17,998 kg
Initial velocity of the train, u = 60.0 km/hr = 60.0 * (1000/3600) m/s (converting km/hr to m/s)
Let v be the velocity at which Superman flies into the train to stop it. The total momentum before the collision is (m1 + m2) * u, and the total momentum after the collision is (m1 + m2) * v. Therefore, we can write:
(m1 + m2) * u = (m1 + m2) * v
Substituting the given values, we have:
(88.0 + 17,998) * (60.0 * (1000/3600)) = (88.0 + 17,998) * v
Now we can solve for v:
v = [(88.0 + 17,998) * (60.0 * (1000/3600))] / (88.0 + 17,998)
v ≈ 165.21 m/s
Therefore, Superman must fly into the train at a speed of approximately 165.21 m/s to stop it.
Now, let's calculate 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.
To calculate the time, we can use Newton's second law, which states that force (F) is equal to mass (m) multiplied by acceleration (a).
We know that the force experienced by the passengers is 0.480 times their own weight. The weight of an object is given by W = mg, where g is the acceleration due to gravity.
Let's assume g ≈ 9.8 m/s².
Using the given force value and the weight equation, we have:
F = 0.480 * mg
Now we can equate the force and mass times acceleration:
ma = 0.480 * mg
The mass on the left side of the equation cancels out, so we are left with:
a = 0.480g
Since Superman is trying to stop the train, the acceleration will be negative.
Now, let's calculate the minimum time using the equation:
v = u + at
Rearranging the equation, we have:
t = (v - u) / -a
Substituting the given values:
t = (0 - 165.21) / (-0.480 * 9.8)
Now we can solve for t:
t ≈ 35.81 seconds
Therefore, the minimum time Superman must take to stop the train is approximately 35.81 seconds.
Next, let's calculate the distance the train travels while being slowed to a stop.
We can use the equations of motion to find the distance traveled. The equation we'll use is:
s = ut + (1/2)at²
In this case, the initial velocity of the train (u) is 60.0 km/hr = 60.0 * (1000/3600) m/s, acceleration (a) is -0.480g, and time (t) is 35.81 seconds.
Now we can calculate the distance:
s = (60.0 * (1000/3600)) * 35.81 + (1/2)(-0.480g)(35.81)²
Substituting the value of g and calculating:
s ≈ 223.86 meters
Therefore, the train travels approximately 223.86 meters while being slowed to a stop.