A 170 pound man has to slide down a rope that can only support 150 pounds. How long will it take him to reach the ground 35 feet below if he keeps the rope from breaking?
To determine the time it takes for the man to reach the ground while considering the weight limit of the rope, we need to calculate the force exerted on the rope by the man.
First, we need to convert the weight of the man from pounds to Newtons. We can do this by multiplying the weight in pounds by the gravitational acceleration, which is approximately 9.8 m/s^2.
Weight of the man = 170 pounds * 9.8 m/s^2 = 1666 Newtons (rounded to the nearest whole number)
Next, we can calculate the force exerted on the rope by the man by subtracting the weight limit of the rope from the weight of the man.
Force exerted on the rope = Weight of the man - Weight limit of the rope
= 1666 Newtons - 150 pounds * 9.8 m/s^2 = 1666 Newtons - 1470 Newtons = 196 Newtons
Now we can use Newton's second law of motion which states that force equals mass multiplied by acceleration. Since the man is sliding down the rope, the force exerted on the rope is equal to the product of the man's mass and the acceleration due to gravity.
Force exerted on the rope = Mass of the man * Acceleration due to gravity
= Mass of the man * 9.8 m/s^2
We can rearrange this equation to solve for the mass of the man:
Mass of the man = Force exerted on the rope / Acceleration due to gravity
= 196 Newtons / 9.8 m/s^2
= 20 kilograms (rounded to the nearest whole number)
Now we can calculate the time it takes for the man to reach the ground using the equation for distance traveled (s) under constant acceleration (a):
s = ut + (1/2) * a * t^2
Where:
s = distance traveled (35 feet = 10.67 meters)
u = initial velocity (0, since the man starts at rest)
a = acceleration due to gravity (9.8 m/s^2)
t = time taken
Rearranging the equation, we have:
t = sqrt((2 * s) / a)
Substituting the given values, we get:
t = sqrt((2 * 10.67 meters) / 9.8 m/s^2)
≈ sqrt(2.18 seconds^2)
≈ 1.48 seconds (rounded to two decimal places)
Therefore, it will take approximately 1.48 seconds for the 170-pound man to reach the ground if he keeps the rope from breaking.