Question
A pilot drops a weight from a plane flying low and slow, taking 3.2 seconds to hit the ground and traveling 120 meters horizontally. If the pilot flies at the same height but twice the speed, how long will it take for the weight to hit the ground and how far will it travel?
Answers
GPT-4o mini
To solve this problem, we first need to analyze the initial scenario where a weight is dropped from a plane flying at a certain speed. The key points are as follows:
1. The weight takes 3.2 seconds to hit the ground.
2. The horizontal distance covered during this time is 120 meters.
3. We need to determine the time it takes for the weight to hit the ground when the plane flies at twice the speed and how far it travels in that time.
### Step 1: Calculate the initial horizontal speed
The horizontal speed of the plane in the initial scenario can be calculated using the formula:
\[
\text{Horizontal Speed} = \frac{\text{Horizontal Distance}}{\text{Time}} = \frac{120 \text{ meters}}{3.2 \text{ seconds}} = 37.5 \text{ m/s}
\]
### Step 2: Analyze the new scenario with twice the speed
When the pilot flies at twice the speed, the new horizontal speed of the plane becomes:
\[
\text{New Horizontal Speed} = 2 \times 37.5 \text{ m/s} = 75 \text{ m/s}
\]
### Step 3: Time to hit the ground
The time it takes for the weight to hit the ground is determined by the vertical motion, which is independent of the horizontal speed. Since the weight is dropped and it took 3.2 seconds to fall from the same height, it will also take 3.2 seconds to hit the ground in the new scenario as the height from which it is dropped does not change.
### Step 4: Calculate the horizontal distance in the new scenario
Now we calculate how far the weight will travel horizontally while it falls:
\[
\text{Horizontal Distance} = \text{Horizontal Speed} \times \text{Time} = 75 \text{ m/s} \times 3.2 \text{ s} = 240 \text{ meters}
\]
### Conclusion
In conclusion, when the pilot flies at twice the speed:
- It will take **3.2 seconds** for the weight to hit the ground.
- The weight will travel **240 meters** horizontally before hitting the ground.
1. The weight takes 3.2 seconds to hit the ground.
2. The horizontal distance covered during this time is 120 meters.
3. We need to determine the time it takes for the weight to hit the ground when the plane flies at twice the speed and how far it travels in that time.
### Step 1: Calculate the initial horizontal speed
The horizontal speed of the plane in the initial scenario can be calculated using the formula:
\[
\text{Horizontal Speed} = \frac{\text{Horizontal Distance}}{\text{Time}} = \frac{120 \text{ meters}}{3.2 \text{ seconds}} = 37.5 \text{ m/s}
\]
### Step 2: Analyze the new scenario with twice the speed
When the pilot flies at twice the speed, the new horizontal speed of the plane becomes:
\[
\text{New Horizontal Speed} = 2 \times 37.5 \text{ m/s} = 75 \text{ m/s}
\]
### Step 3: Time to hit the ground
The time it takes for the weight to hit the ground is determined by the vertical motion, which is independent of the horizontal speed. Since the weight is dropped and it took 3.2 seconds to fall from the same height, it will also take 3.2 seconds to hit the ground in the new scenario as the height from which it is dropped does not change.
### Step 4: Calculate the horizontal distance in the new scenario
Now we calculate how far the weight will travel horizontally while it falls:
\[
\text{Horizontal Distance} = \text{Horizontal Speed} \times \text{Time} = 75 \text{ m/s} \times 3.2 \text{ s} = 240 \text{ meters}
\]
### Conclusion
In conclusion, when the pilot flies at twice the speed:
- It will take **3.2 seconds** for the weight to hit the ground.
- The weight will travel **240 meters** horizontally before hitting the ground.