At the train station, you notice a large horizontal spring at the end of the track where the train comes in. This is a safety device to stop the train so that it will not plow through the station if the engineer misjudges the stopping distance. While waiting, you wonder what would be the fastest train that the spring could stop at its full compression which is L=2ft . To keep the passengers safe when the train stops, you assume a maximum stopping acceleration of g/2. You also guess that a train weighs half a million lbs. For purpose of getting an estimate, you decide to assume that all frictional force are negligible.
vf^2=vi^2+2ad
vi=sqrt(2gd/2=sqrt(gd)
if g=32ft/s^2 and d=2ft, vi will be in ft/sec
To determine the speed at which the spring could stop the train at full compression, we can use the principle of conservation of mechanical energy.
First, let's calculate the gravitational potential energy (PE) of the train at the top of the compression, when it is at rest:
PE = mgh
Where:
m = mass of the train = 500,000 lbs = 250,000 slugs (1 slug is the unit of mass in the English system)
g = acceleration due to gravity = 32.2 ft/s^2 (approx.)
PE = 250,000 slugs * 32.2 ft/s^2 * 2 ft
PE = 16,100,000 ft-lbs
Next, let's calculate the total energy of the spring at full compression:
E = 0.5kL^2
Where:
k = spring constant (stiffness)
L = full compression distance = 2 ft
Since we don't have the exact spring constant, we need to rearrange the equation to solve for k:
k = 2E / L^2
Let's assume the maximum stopping acceleration is g/2, which means the deceleration is -g/2. Therefore, the force exerted by the spring can be calculated using Newton's second law:
F = -ma
F = -m(-g/2)
F = mg/2
At maximum compression, the force exerted by the spring is equal to the weight of the train:
F = mg/2 = 250,000 slugs * 32.2 ft/s^2 / 2
F = 4,025,000 ft-lbs
Now, let's equate the force exerted by the spring to the spring constant times the compression distance:
F = kL
4,025,000 ft-lbs = k * 2 ft
Therefore, we can solve for k:
k = 4,025,000 ft-lbs / 2 ft
k = 2,012,500 lbs
Finally, substitute the obtained value of k back into the equation for total energy of the spring:
E = 0.5 * 2,012,500 lbs * (2 ft)^2
E = 4,050,000 ft-lbs
Now we can calculate the initial kinetic energy (KE) of the train using the principle of conservation of mechanical energy:
KE + PE = E
KE = E - PE
KE = 4,050,000 ft-lbs - 16,100,000 ft-lbs
KE = - 12,050,000 ft-lbs
Since kinetic energy cannot be negative, this means the train cannot be accelerated to full compression using only the spring. Therefore, we cannot determine the speed at which the spring could stop the train at its full compression distance.
To determine the fastest train that the spring could stop at its full compression, we can make use of the work-energy principle.
The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by the spring is equal to the change in kinetic energy of the train.
The work done by the spring can be calculated using Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.
Let's break down the steps to find the answer:
Step 1: Convert the weight of the train from pounds to newtons.
- Given that the train weighs half a million pounds, we need to convert this to newtons using the conversion factor 1 lb = 4.44822 N.
Weight of the train = 500,000 lbs * 4.44822 N/lb
Step 2: Calculate the force exerted on the spring.
- Since the train comes to a stop, its initial velocity is assumed to be some maximum velocity, V_max.
- The force exerted by the spring is equal to the mass of the train multiplied by the maximum stopping acceleration.
Force exerted by the spring = Weight of the train * Maximum stopping acceleration
Step 3: Calculate the work done by the spring.
- The work done by the spring can be calculated as the force exerted by the spring multiplied by the distance of spring compression, which is given as L=2 feet.
Work done by the spring = Force exerted by the spring * Distance of spring compression
Step 4: Relate the work done by the spring to the change in kinetic energy.
- According to the work-energy principle, the work done by the spring is equal to the change in kinetic energy of the train.
- The change in kinetic energy is equal to the difference between the final kinetic energy (when the train comes to a stop) and the initial kinetic energy (when the train was traveling at its maximum velocity).
Change in kinetic energy = Final kinetic energy - Initial kinetic energy
Step 5: Solve for the final kinetic energy.
- We can assume that when the train comes to a stop, its final velocity is zero.
- The final kinetic energy is given by the equation: final kinetic energy = (1/2) * mass * (final velocity)^2.
Step 6: Calculate the maximum velocity of the train.
- To find the fastest train that the spring can stop, we need to find the maximum velocity of the train before it comes to a stop.
- Rearranging the equation in step 5, we can solve for the final velocity.
Step 7: Calculate the fastest train that the spring can stop.
- Now that we know the maximum velocity, we can determine the fastest train that can be stopped by the spring.
With the given steps, we can determine the fastest train that the spring could stop at its full compression of L=2 feet.