A truck is traveling at a speed of 33.1 m/s along a level road. A crate is resting on the bed of the truck, and the coefficient of static friction between the crate and the truck bed is 0.632. Determine the shortest distance in which the truck can come to a halt without causing the crate to slip forward relative to the truck.

To determine the shortest distance in which the truck can come to a halt without causing the crate to slip forward relative to the truck, we need to consider the maximum static friction force between the crate and the truck bed.

The formula for static friction is given by:

Fs = μs * N

where Fs is the static friction force, μs is the coefficient of static friction, and N is the normal force.

The normal force N is equal to the weight of the crate, which is given by:

N = m * g

where m is the mass of the crate and g is the acceleration due to gravity.

Assuming the truck is moving with a constant velocity, the net force acting on the crate in the horizontal direction is zero.

The force of static friction exerted on the crate in the opposite direction of motion is equal to the forward force exerted on the crate, which is given by:

F = m * a

where m is the mass of the crate and a is the acceleration of the truck.

Setting these two forces equal to each other, we have:

Fs = F
μs * N = m * a

Substituting the expression for N, we get:

μs * m * g = m * a

Simplifying, we find:

μs * g = a

Now, we can calculate the acceleration of the truck:

a = μs * g = 0.632 * 9.8 m/s^2 = 6.1176 m/s^2

To find the shortest stopping distance, we can use the kinematic equation:

v^2 = u^2 + 2as

where v is the final velocity (which is 0 m/s since the truck comes to a halt), u is the initial velocity (which is 33.1 m/s), a is the acceleration, and s is the stopping distance.

Rearranging the equation:

s = (v^2 - u^2) / (2a)

Substituting the values:

s = (0^2 - 33.1^2) / (2 * -6.1176)
s = 557.83 meters

Therefore, the shortest distance in which the truck can come to a halt without causing the crate to slip forward relative to the truck is 557.83 meters.