The tops of the towers of the Golden Gate Bridge, in San Francisco, are 227m above the water. Suppose a worker drops a wrench from the top of a tower. If the average force of air resistance is 22% of the force of free fall, with what speed will the wrench hit the water?

To determine the speed at which the wrench hits the water, we can use the principle of free fall and account for the force of air resistance. Here's how you can calculate it:

1. Start by understanding the concept of free fall. In a vacuum or when air resistance is negligible, any object dropped from a height will fall with a constant acceleration due to gravity. The acceleration due to gravity on Earth is approximately 9.8 m/s².

2. Determine the time it takes for the wrench to fall from the top of the tower to the water. To do this, you can use the equation for distance traveled during free fall: s = (1/2)gt², where s is the distance, g is the acceleration due to gravity, and t is the time in seconds.

3. Substitute the given height into the distance equation: 227m = (1/2)(9.8 m/s²)t².

4. Rearrange the equation to solve for t: t = √(2s/g). Plug in the values: t = √(2 * 227m / 9.8 m/s²).

5. Calculate the time it takes for the wrench to fall: t ≈ 6.02 seconds.

6. Now, we need to consider the force of air resistance. According to the problem, the average force of air resistance is 22% of the force of free fall.

7. Calculate the force of free fall by multiplying the mass of the wrench by the acceleration due to gravity: F_free_fall = mg, where m is the mass in kilograms.

8. Calculate the force of air resistance: F_air_resistance = 0.22 * F_free_fall.

9. By Newton's second law of motion (F = ma), the force of air resistance is equal to mass times acceleration. Rearrange the formula to find acceleration: a = F_air_resistance / m.

10. Substitute the values and solve for acceleration: a = (0.22 * F_free_fall) / m.

11. Now, we need to determine the final velocity of the wrench just before it hits the water. Use the equation: v = u + at, where v is the final velocity, u is the initial velocity (which is 0 because the wrench is dropped vertically downward), a is the acceleration, and t is the time.

12. Plug the values into the equation and solve for v: v = 0 + a * t.

13. Calculate the speed at which the wrench hits the water by taking the absolute value of the final velocity.

By following these steps, you should be able to calculate the speed at which the wrench will hit the water while accounting for the force of air resistance.

To determine the speed at which the wrench will hit the water, we need to calculate the initial velocity of the wrench when it is dropped from the top of the tower.

We can start by finding the gravitational force acting on the wrench. The force of gravity can be calculated using the equation:

F = mg

where F is the gravitational force, m is the mass of the wrench, and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Since the mass of the wrench is not given, we can assume a typical value of 0.5 kg.

F = (0.5 kg)(9.8 m/s^2)
F ≈ 4.9 N

Next, we need to determine the force of air resistance. According to the problem, the average force of air resistance is 22% of the force of free fall. Therefore, the force of air resistance can be calculated as:

Fa = 0.22F

Fa ≈ (0.22)(4.9 N)
Fa ≈ 1.078 N

Now, we can calculate the net force acting on the wrench:

Net force = F - Fa
Net force = 4.9 N - 1.078 N
Net force ≈ 3.822 N

Using Newton's second law of motion, we can relate the net force to the acceleration of the wrench:

Net force = ma

Solving for acceleration:

a = Net force / m
a ≈ 3.822 N / 0.5 kg
a ≈ 7.644 m/s^2

Now we can use the kinematic equation to calculate the final velocity (v) of the wrench when it hits the water:

v^2 = u^2 + 2as

where u is the initial velocity (which is zero when the wrench is dropped) and s is the distance traveled (227 m).

v^2 = 0 + 2(7.644 m/s^2)(227 m)
v^2 = 2(7.644 m/s^2)(227 m)
v^2 ≈ 3485.96 m^2/s^2

Taking the square root of both sides, we find:

v ≈ √3485.96 m^2/s^2
v ≈ 59.01 m/s

Therefore, the wrench will hit the water with a speed of approximately 59.01 m/s.

78% of the available potential energy will be converted to kinetic energy. The rest will be converted to frictional heating.

0.78 M g H= (M/2) V^2

where V it the velocity when it hits the water and H = 227 m

V = sqrt(1.56 g H) = 58.9 m/s