You hold a hose at 45 degrees
to the horizontal and at a height of 1 m from the floor. The water reaches a
maximum distance of 10 m from where you are standing. Now you place your thumb over the end of
the hose to occlude the opening by 80%, which in turn reduces the flow rate by 50%. Even though less
fluid emerges the water travels further?
a) How far does the water travel with your thumb over the end of the hose (assuming the height and
angle remain the same)?
b) Assume the flow rate (as a percentage of original flow rate) can given by
Q%=100[1-(A/100)^5]
where A% is the percentage by which the opening of the hose is occluded. Notice that if A% is 0% then Q%
is 100%, whereas if A% is 100% then Q% is 0%. Find the amount by which the hose must be
occluded in order for the water to travel twice as far (assuming the height and angle remain the same).
To answer these questions, we need to analyze the given scenario step by step. Let's break it down.
a) How far does the water travel with your thumb over the end of the hose (assuming the height and angle remain the same)?
When your thumb covers 80% of the opening, it means only 20% of the original flow rate is coming out. So, let's calculate the new flow rate using the provided formula:
Q% = 100[1 - (A/100)^5]
Substituting A = 80%:
Q% = 100[1 - (80/100)^5]
= 100[1 - (0.8)^5]
= 100[1 - 0.32768]
= 100(0.67232)
= 67.232
Therefore, the water travels 67.232% of the original maximum distance. To find the actual distance traveled, we need to multiply this percentage with the original maximum distance of 10 m:
Distance traveled = 67.232% of 10 m
= 0.67232 * 10 m
= 6.7232 m
So, with your thumb over the end of the hose, the water will travel approximately 6.7232 meters.
b) Find the amount by which the hose must be occluded in order for the water to travel twice as far (assuming the height and angle remain the same).
Here, we want the water to travel twice as far as the original maximum distance of 10 m. Let's call the occlusion percentage we need to find as B.
Using the same formula for flow rate:
Q% = 100[1 - (B/100)^5]
We want the water to travel twice the distance, so:
Distance traveled = 2 * 10 m
= 20 m
Now we can solve for B by rearranging the formula:
1 - (B/100)^5 = (Distance traveled/100)^5
= (20/100)^5
= 0.2^5
≈ 0.032
Taking the fifth root of both sides:
B/100 = (1 - 0.032)^(1/5)
B/100 ≈ ∛(0.968)^5
B/100 ≈ ∛0.8754709376
B/100 ≈ 0.961
Multiplying both sides by 100:
B ≈ 96.1
Therefore, the hose must be occluded by approximately 96.1% for the water to travel twice as far.
Note: These calculations are based on the given formulas and assumptions. Actual observations may differ due to various factors such as air resistance and other external factors.