A train of mass 200 tonne travelling at 140 km/hour applies the brakes fiercely, so
that the brakes lock and the wheels slide along the rails. It stops after sliding
1000m.
Calculate:
(i) The acceleration,
(ii) The longitudinal force on the rails,
(iii) The coefficient of friction between the rails and the wheels
To calculate the requested values, let's use the following formulas:
(i) The acceleration (a) can be calculated using the formula:
a = (v_f^2 - v_i^2) / (2 * d)
where v_f is the final velocity, v_i is the initial velocity, and d is the distance.
(ii) The longitudinal force (F) on the rails can be calculated using the formula:
F = m * a
where m is the mass of the train.
(iii) The coefficient of friction (μ) between the rails and the wheels can be calculated using the formula:
μ = F / (m * g)
where g is the acceleration due to gravity.
Given:
mass (m) = 200 tonne = 200,000 kg
initial velocity (v_i) = 140 km/h = 140,000 m/3600 s
final velocity (v_f) = 0 m/s (since the train stops)
distance (d) = 1000 m
acceleration due to gravity (g) = 9.8 m/s^2
Let's calculate each value step-by-step:
(i) Acceleration (a):
a = (v_f^2 - v_i^2) / (2 * d)
= (0^2 - (140,000)^2) / (2 * 1000)
= -(-140,000,000,000) / 2,000
= 70,000,000 m/s^2
(ii) Longitudinal force (F):
F = m * a
= 200,000 kg * 70,000,000 m/s^2
= 14,000,000,000,000 N
(iii) Coefficient of friction (μ):
μ = F / (m * g)
= 14,000,000,000,000 N / (200,000 kg * 9.8 m/s^2)
= 7.14
Therefore:
(i) The acceleration is 70,000,000 m/s^2.
(ii) The longitudinal force on the rails is 14,000,000,000,000 N.
(iii) The coefficient of friction between the rails and the wheels is 7.14.
To solve this problem, we will use the following equations of motion:
(i) Acceleration (a) = (Final velocity - Initial velocity) / Time
(ii) Longitudinal force (F) = Mass * Acceleration
(iii) Coefficient of friction (μ) = Longitudinal force / Normal force
First, let's convert the mass of the train from tonnes to kilograms:
Mass = 200 tonnes * 1000 kg/tonne = 200,000 kg
Now, we need to convert the initial velocity from km/h to m/s:
Initial velocity = 140 km/h * (1000 m/1 km) * (1 h/3600 s) = 38.9 m/s
Since the brakes lock and the wheels slide along the rails, the final velocity is 0 m/s. The time it takes to stop can be calculated using the equation:
Final velocity = Initial velocity + (Acceleration * Time)
0 = 38.9 + (Acceleration * Time)
As we slide for 1000 m, we can use the equation of motion to find the time it takes to stop:
Distance = Initial velocity * Time + (1/2) * Acceleration * Time^2
1000 = 38.9 * Time + (1/2) * Acceleration * Time^2
Now, we have two equations with two unknowns (Acceleration and Time). We can solve for both of these unknowns simultaneously.
First, solve the equation 0 = 38.9 + (Acceleration * Time) for Time:
Time = -38.9 / Acceleration
Next, substitute this value of Time into the equation 1000 = 38.9 * Time + (1/2) * Acceleration * Time^2:
1000 = 38.9 * (-38.9 / Acceleration) + (1/2) * Acceleration * (-38.9 / Acceleration)^2
Simplify the equation:
1000 = -38.9^2 / Acceleration + (1/2) * (-38.9^2 / Acceleration)
Multiply both sides by Acceleration:
1000 * Acceleration = -38.9^2 + (1/2) * (-38.9^2)
Combine the terms:
1000 * Acceleration = -(1 + 1/2) * (38.9^2)
Solve for Acceleration:
Acceleration = -(1 + 1/2) * (38.9^2) / 1000
After performing the calculations, we find:
Acceleration ≈ -11.36 m/s^2
Next, using this acceleration, we can find the longitudinal force on the rails:
Force = Mass * Acceleration
Force = 200,000 kg * (-11.36 m/s^2)
After performing the calculations, we find:
Force ≈ -2,272,000 N
Finally, we can find the coefficient of friction between the rails and the wheels:
Coefficient of friction = Longitudinal force / Normal force
To find the normal force, we need to consider that when the brakes lock, the vertical force on the wheels due to gravity (equal to the weight of the train) is balanced by the normal force from the rails:
Normal force = Weight of the train = Mass * Gravity
Where gravity ≈ 9.8 m/s^2.
Normal force = 200,000 kg * 9.8 m/s^2
After performing the calculation, we find:
Normal force ≈ 1,960,000 N
Now we can calculate the coefficient of friction:
Coefficient of friction = Longitudinal force / Normal force
Coefficient of friction = -2,272,000 N / 1,960,000 N
After performing the calculation, we find:
Coefficient of friction ≈ -1.16
Please note that the negative value for acceleration and coefficient of friction indicates the opposing direction compared to the initial motion of the train.