A 67.0 kg diver steps off a diving board and drops straight down into the water. The water provides an upward average net force 1480 N. If the diver comes to rest 4.7 m below the water's surface, what is the total distance between the diving board and the diver's stopping point underwater?
the upward acceleration from the water is a=F/m = 1480/67 = 22.09 m/s^2
The diver's speed on entering the water is
v = √(2as) = √(2*22.09*4.7) = 14.4 m/s
In the air, the diver's speed is
v = 4.9t^2, so he fell for
t = √(14.4/4.9) = 1.71 seconds
The diver's height upon diving is
h = Ho - 4.9t^2
h=0 at t=1.71, so
Ho = 4.9*1.71^2 = 14.3
The total distance is thus 14.3+4.7 = 19.0 meters
the diver's air speed is v = 9.8t
make that correction and redo the board height.
To find the total distance between the diving board and the diver's stopping point underwater, we need to consider two components: the distance the diver falls before reaching the water's surface and the distance the diver continues to sink below the water's surface after coming to rest.
Let's break down the problem step by step:
Step 1: Calculate the distance the diver falls before reaching the water's surface.
We can use the equations of motion to determine this distance. Since the diver is in free fall, we can use the equation:
s = ut + (1/2)gt^2
Where:
- s is the distance fallen
- u is the initial velocity (which is 0 since the diver starts from rest)
- t is the time taken to reach the water's surface
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
In this case, the distance fallen is equal to the height of the diving board. However, be sure to convert this height into meters if it's in a different unit. Let's assume the height of the diving board is 10 meters:
s = 0 + (0.5)(9.8)t^2
Since the diver dropped straight down, we can ignore the initial velocity (u = 0).
We also know that the distance fallen is equal to the height of the diving board, which is 10 meters.
10 = (0.5)(9.8)t^2
Now solve for t. Rearrange the equation:
t^2 = (2 * 10) / 9.8
t^2 = 20 / 9.8
t^2 ≈ 2.0408
Taking the square root of both sides:
t ≈ √2.0408
t ≈ 1.429 seconds
Therefore, the diver reaches the water's surface after approximately 1.429 seconds.
The distance fallen before reaching the water's surface can now be calculated by substituting the value of time (t) into the equation:
s = (0)(1.429) + (0.5)(9.8)(1.429)^2
s ≈ 10 meters
So, the distance fallen before reaching the water's surface is approximately 10 meters.
Step 2: Calculate the distance the diver continues to sink below the water's surface after coming to rest.
The average net upward force provided by the water is 1480 N. This force is equal to the weight of the diver, which can be calculated using the equation:
F = mg
Where:
- F is the force acting on the diver (1480 N)
- m is the mass of the diver (67.0 kg)
- g is the acceleration due to gravity (9.8 m/s^2)
1480 = 67.0 * 9.8
Now solve for mass (m):
m = 1480 / 9.8
m ≈ 151.02 kg
The weight of the diver is approximately 151.02 kg.
To calculate the distance the diver continues to sink below the water's surface after coming to rest, we can use the equation:
s = (F / buoyant force) * (density of water / g)
Where:
- s is the distance the diver sinks below the water's surface
- F is the net upward force exerted by the water (1480 N)
- buoyant force is equal to the weight of the water displaced by the diver (which is equal to the weight of the diver, since the diver is at rest)
- density of water is approximately 1000 kg/m^3 (density of water)
- g is the acceleration due to gravity (9.8 m/s^2)
Substituting the values:
s = (1480 / 151.02) * (151.02 / 1000) * (1 / 9.8)
Now, calculate:
s ≈ (9.797) * (0.15102) * (0.1020408)
s ≈ 0.15092 meters
Therefore, the distance the diver continues to sink below the water's surface after coming to rest is approximately 0.15092 meters.
Step 3: Find the total distance between the diving board and the diver's stopping point underwater.
To find the total distance, simply add the distance fallen before reaching the water's surface to the distance the diver continues to sink below the water's surface:
Total distance = Distance fallen + Distance sunk
Total distance = 10 meters + 0.15092 meters
Total distance ≈ 10.15 meters
Therefore, the total distance between the diving board and the diver's stopping point underwater is approximately 10.15 meters.