A flatbed egg truck is in a drag race. The truck is carrying a 450 lb. crate of eggs in the back. The coefficients of friction between the crate and the truck bed are μs = 0.9 and μk = 0.4. What is the maximum acceleration the truck can have without the eggs sliding off the back of the truck?

m•a=F(fr)=μ(s)•m•g

a= μ(s)•g

To determine the maximum acceleration the truck can have without the eggs sliding off the back, we need to consider the forces acting on the crate of eggs.

First, let's analyze the situation when the crate is at rest. In this case, the static friction force acts between the crate and the truck bed to prevent it from sliding. The maximum value for static friction can be calculated using the formula:

Fs_max = μs * N

Where:
- Fs_max is the maximum static friction force
- μs is the coefficient of static friction
- N is the normal force acting on the crate

The normal force N is equal to the weight of the crate, which is given as 450 lb. Converting this to pounds-force (lbf) by multiplying by the acceleration due to gravity (32.2 ft/s^2), we get:

N = 450 lbf * 32.2 ft/s^2 = 14,535 lbf·ft/s^2

Now we can calculate the maximum static friction force:

Fs_max = 0.9 * 14,535 lbf·ft/s^2 = 13,082 lbf·ft/s^2

The static friction force will oppose any force that tries to move the crate. As long as the truck's acceleration remains lower than 13,082 lbf·ft/s^2 (or its equivalent in other units), the eggs will not slide off.

However, since we are looking for the maximum acceleration, we need to consider the case where the crate is in motion. In this scenario, the kinetic friction force is relevant.

The kinetic friction force between the crate and the truck bed can be calculated using the formula:

Fk = μk * N

Where:
- Fk is the kinetic friction force
- μk is the coefficient of kinetic friction
- N is the normal force (weight of the crate)

Using the same value for N as before, we can now calculate the maximum kinetic friction force:

Fk_max = 0.4 * 14,535 lbf·ft/s^2 = 5,814 lbf·ft/s^2

The maximum acceleration of the truck without the eggs sliding off is equal to the maximum kinetic friction force divided by the mass of the crate.

Given that the weight of the crate is 450 lb, we need to convert it to mass by dividing by the acceleration due to gravity:

Mass = 450 lb / 32.2 ft/s^2 = 13.98 slug (approximately)

Now we can calculate the maximum acceleration:

Acceleration_max = Fk_max / Mass

Acceleration_max = (5,814 lbf·ft/s^2) / 13.98 slug

Acceleration_max ≈ 416.33 ft/s^2

Therefore, the maximum acceleration the truck can have without the eggs sliding off the back is approximately 416.33 ft/s^2.