A large box is loaded into the very back of a pickup truck and is not tied down. The truck then goes down the road at 20.0 m/s. The coefficient of static friction is 0.80 and the coefficient of kinetic friction is 0.45 between the box and the bed of the truck. What is the shortest distance that the truck can stop without the box sliding?

To determine the shortest distance that the truck can stop without the box sliding, we need to consider the forces acting on the box and calculate the maximum static friction force that can prevent it from sliding.

Let's break this down step by step:

1. Identify the forces acting on the box:
- Gravity (mg): The weight of the box pulling it downward.
- Normal force (N): The force exerted by the truck bed on the box, perpendicular to the surface.
- Static friction force (fs): The force opposing the box's tendency to slide.
- Kinetic friction force (fk): The force opposing the motion of the box when it is sliding.

2. Calculate the weight of the box (mg):
- The weight of an object is given by the formula: weight = mass x acceleration due to gravity.
- Since the mass of the box is not given, we can't calculate the weight directly. However, we can assume a mass for the box (let's say 1 kg) to simplify the calculation.

weight = 1 kg x 9.8 m/s^2 = 9.8 N (downward)

3. Calculate the normal force (N):
- The normal force is equal in magnitude but opposite in direction to the weight of the box.
- Therefore, the normal force is 9.8 N (upward).

4. Calculate the maximum static friction force (fs):
- The maximum static friction force is given by the formula: fs = coefficient of static friction x normal force.
- In this case, the coefficient of static friction is 0.80 and the normal force is 9.8 N.

fs = 0.80 x 9.8 N = 7.84 N

5. Determine if the maximum static friction force can stop the box:
- Since the box is not sliding, the static friction force must be equal to or greater than the force trying to slide the box (in this case, the weight of the box).

fs ≥ mg
7.84 N ≥ 9.8 N

- Since 7.84 N is less than 9.8 N, the maximum static friction force is not enough to prevent the box from sliding.

6. Calculate the acceleration of the box when sliding (a):
- The acceleration of the box is given by the formula: a = fk / mass.
- Since the mass is not given, let's assume it is still 1 kg.

a = fk / 1 kg

7. Calculate the kinetic friction force (fk):
- The kinetic friction force is given by the formula: fk = coefficient of kinetic friction x normal force.
- In this case, the coefficient of kinetic friction is 0.45 and the normal force is 9.8 N.

fk = 0.45 x 9.8 N = 4.41 N

8. Calculate the acceleration (a) using fk:
- Using Newton's second law of motion: a = fk / mass.
- Since the mass is still assumed to be 1 kg, the acceleration is equal to the kinetic friction force (a = fk = 4.41 m/s^2).

9. Calculate the distance (d):
- We will use the equation of motion: v^2 = u^2 + 2ad, where u = initial velocity (20.0 m/s), v = final velocity (0 m/s), and a = acceleration (-4.41 m/s^2).

0^2 = (20.0 m/s)^2 + 2(-4.41 m/s^2)d

- Solving for d:

0 = 400 - 8.82d
8.82d = 400
d = 400 / 8.82 ≈ 45.33 m

Therefore, the shortest distance that the truck can stop without the box sliding is approximately 45.33 meters.