The floor of a railroad flatcar is loaded with loose crates having a coefficient of static friction of 0.20 with the floor. If the train is initially moving at a speed of 47 km/h, in how short a distance can the train be stopped at constant acceleration without causing the crates to slide over the floor?

47 km/h = 13.056 m/s

Compute the acceleration(a)when the stopping distance is X meters:

V^2 = 170.44 m^2/s^2 = 2 a X
a = V^2/(2X) = 85.22/X

M g * us = M a when a is as high as possible without slipping

M's cancel. You know the static fricion coefficient us. Substitute for a and solve for X

87m

coefficient of static friction = 0.20

fs = static frictional force
mass = m
g = 9.8m/s^2
as we know that fs = fnormal*coefficient of static friction
so, fs= mg*0.20
as fnormal= mg = 9.8m N
now, acc to newtons law
fnet = fs
ma = 9.8*0.20*m
m's get cancel
a= 1.96m/s^2
applying the formula
v^2= u^2 + 2ax
now substitute the value in this equation
you get, x= 43.1m
don't forget to change speed into m/s

To determine the distance required to stop the train without causing the crates to slide over the floor, we need to calculate the maximum acceleration the train can have before the crates start sliding. Once we have the acceleration, we can use the kinematic equation to find the distance.

Here's how to solve the problem step by step:

Step 1: Convert the speed of the train from km/h to m/s.
- Divide the speed by 3.6 since there are 3.6 meters in one second.
- 47 km/h ÷ 3.6 = 13.06 m/s (rounded to the nearest hundredth).

Step 2: Determine the maximum acceleration that the train can have.
- The maximum acceleration is the one that overcomes the static friction and causes the crates to start sliding.
- The formula to calculate the maximum acceleration is:
a_max = μ_s * g
where μ_s is the coefficient of static friction and g is the acceleration due to gravity (approximately 9.8 m/s²).
- Substitute the values:
a_max = 0.20 * 9.8 = 1.96 m/s² (rounded to the nearest hundredth).

Step 3: Calculate the stopping distance using the kinematic equation.
- The formula to calculate the stopping distance is:
d = v² / (2 * a)
where d is the stopping distance, v is the initial velocity, and a is the acceleration.
- Substitute the values:
d = (13.06 m/s)² / (2 * 1.96 m/s²)
d = 170.35 m²/s² / 3.92 m/s²
d = 43.47 meters (rounded to the nearest hundredth).

Therefore, the train can be stopped in approximately 43.47 meters without causing the crates to slide over the floor.