A flatbed truck of mass 2000kg traveling 20 m/s (about 45mph)carries a load of mass 500kg positioned 3 meters behind the cab. The load is kept on the truck by friction, and the coefficient of friction between the load and the bed of the truck is 0.5. What is the shortest distance in which the truck can stop without having the load slide forward enough to hit the cab? (Note that the load can slide forward 3 meters before it hits the cab.)Please show work.Help!
shyro
To find the shortest distance in which the truck can stop without the load sliding forward enough to hit the cab, we first need to find the maximum force of static friction between the load and the bed of the truck.
The maximum force of static friction can be calculated using the formula:
F_friction = coefficient of friction * normal force
The normal force is equal to the weight of the load, which is given by:
Normal force = mass * gravity
where mass is the mass of the load and gravity is the acceleration due to gravity (approximately 9.8 m/s²).
So, the normal force can be calculated as:
Normal force = 500 kg * 9.8 m/s² = 4900 N
Now, we can calculate the maximum force of static friction:
F_friction = 0.5 * 4900 N = 2450 N
The maximum force of static friction is 2450 N, which means that the load can exert this force on the truck bed in the opposite direction of motion before it starts sliding forward.
To stop the load from sliding forward, the deceleration of the truck must be at least equal to the acceleration due to gravity.
The deceleration of the truck can be calculated using the formula:
Deceleration = (final velocity - initial velocity) / time taken
Since we want to find the shortest distance, we can assume that the final velocity is zero (the truck comes to a complete stop). We have the initial velocity (20 m/s) and we need to find the time taken.
The time taken can be calculated using the formula:
Distance = (initial velocity + final velocity) / 2 * time taken
Here, distance is the total distance covered by the truck and load before stopping. The initial velocity is 20 m/s and the final velocity is zero.
Plugging in the values, we get:
Distance = (20 m/s + 0) / 2 * time taken
Distance = 10 m/s * time taken
To find the time taken, we need to find the deceleration first.
Deceleration = (final velocity - initial velocity) / time taken
Since the final velocity is zero, the deceleration becomes:
Deceleration = -initial velocity / time taken
We know that the deceleration must be equal to the acceleration due to gravity:
9.8 m/s² = -20 m/s / time taken
Solving for time taken:
time taken = -20 m/s / 9.8 m/s²
time taken = -2.04 s
Since time cannot be negative, we take the positive value of time taken:
time taken = 2.04 s
Now, we can calculate the distance:
Distance = 10 m/s * 2.04 s
Distance = 20.4 meters
Therefore, the shortest distance in which the truck can stop without having the load slide forward enough to hit the cab is approximately 20.4 meters.