Starting from rest, a 98-kg firefighter slides down a fire pole. The average frictional force everted on him by the pole has a magnitude of 700 N, and his speed at the bottom of the pole is 3.4 m/s. How far did he slide down the pole?
Why did the firefighter slide down the pole? Because he wanted to pole-vault his way into the fire, of course! *ba dum tss* Now, let's get down to business.
We can use the work-energy principle to solve this problem. The work done by the frictional force will be equal to the change in the firefighter's kinetic energy.
The work done by the frictional force is given by:
Work = Force × Distance × cos(θ)
Since the firefighter slides straight down the pole without deviating from the vertical path, the angle between the displacement and the frictional force is 0 degrees. Therefore, cos(θ) = cos(0) = 1.
The work done by the friction force is also equal to the change in the firefighter's kinetic energy:
Work = Change in kinetic energy = (1/2) × Mass × (Final velocity^2 - Initial velocity^2)
We can rearrange this equation to solve for the distance:
Distance = Work / (Force × cos(θ))
Plugging in the given values:
Distance = (Change in kinetic energy) / (Force × cos(θ))
= [(1/2) × Mass × (Final velocity^2 - Initial velocity^2)] / (Force × cos(θ))
= [(1/2) × 98 kg × (3.4 m/s)^2 - 0] / 700 N
= 30.7 m
So, the firefighter slides down the pole for approximately 30.7 meters. Phew, that's quite a ride!
To find the distance the firefighter slid down the pole, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy.
In this case, the work done by the frictional force on the firefighter will be equal to the change in his kinetic energy. The work done by a force is given by the equation:
Work = Force * Distance
The frictional force acting on the firefighter is equal to the product of the average frictional force and the distance traveled down the pole:
Frictional force = Average frictional force * Distance
Since the firefighter starts from rest, his initial kinetic energy is zero. The final kinetic energy is given by the equation:
Final kinetic energy = (1/2) * mass * (final velocity)^2
The work done by the frictional force is equal to the change in kinetic energy:
Work = Final kinetic energy - Initial kinetic energy
Therefore, we have:
Average frictional force * Distance = (1/2) * mass * (final velocity)^2 - 0
Simplifying this equation, we get:
Distance = [(1/2) * mass * (final velocity)^2] / Average frictional force
Plugging in the given values:
Mass = 98 kg
Final velocity = 3.4 m/s
Average frictional force = 700 N
Distance = [(1/2) * 98 kg * (3.4 m/s)^2] / 700 N
Calculating this expression, we find:
Distance = 5.1 m
Therefore, the firefighter slid down the pole a distance of 5.1 meters.
To find the distance the firefighter slid down the pole, we can use the work-energy principle. The work done by the frictional force on the firefighter is equal to the change in his kinetic energy.
The work done by a force is given by the formula: work = force * distance * cos(angle)
In this case, the angle between the force and the direction of motion is 0 degrees because the force is acting directly opposite to the motion. Therefore, the cosine of 0 degrees is 1, so we can simplify the formula to: work = force * distance.
The work done by the frictional force is negative because it acts opposite to the direction of motion. So, the equation becomes: -force * distance = change in kinetic energy.
The change in kinetic energy is given by the formula: change in kinetic energy = 1/2 * mass * (final velocity)^2 - 1/2 * mass * (initial velocity)^2.
In this case, the initial velocity is 0 m/s because the firefighter starts from rest. So, the equation becomes: -force * distance = 1/2 * mass * (final velocity)^2.
Substituting the given values into the equation, we get: -700 N * distance = 1/2 * 98 kg * (3.4 m/s)^2.
Simplifying the equation, we get: -700 N * distance = 1/2 * 98 kg * 11.56 m^2/s^2.
Now, we can solve for the distance by isolating it in the equation: distance = (1/2 * 98 kg * 11.56 m^2/s^2) / -700 N.
Evaluating the expression, we find: distance ≈ -1.03 m.
Since distance cannot be negative in this context, we assume a positive value. Therefore, the firefighter slid down approximately 1.03 meters.