A hare sees a dog 200 m away from her and scuds off in the opposite direction at a speed of 24 km/hr. Two minutes later, the dog perceives her and gives chase at a speed of 32 km/hr. How soon will the dog overtake the hare and what is the distance from the spot from where the hare took flight?

Options

A. 8 min 2 km
B. `7(1)/(2)` min, 2 km
C. `7(1)/(2)` min, 3 km
D. `7(1)/(2)` min, 1 km

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Convert the hare's speed from km/hr to m/s
The hare's speed is given as 24 km/hr. To convert this to meters per second (m/s), we use the conversion factor:
\[ 1 \text{ km/hr} = \frac{1000 \text{ m}}{3600 \text{ s}} = \frac{1}{3.6} \text{ m/s} \]
So,
\[ \text{Speed of hare} = 24 \text{ km/hr} \times \frac{1}{3.6} \text{ m/s} = \frac{24}{3.6} \text{ m/s} = 6.67 \text{ m/s} \]

### Step 2: Calculate the distance covered by the hare in 2 minutes
The dog starts chasing the hare 2 minutes later. We need to find out how far the hare has run in that time.
\[ \text{Time in seconds} = 2 \text{ minutes} \times 60 \text{ seconds/minute} = 120 \text{ seconds} \]
Now, using the speed of the hare:
\[ \text{Distance covered by hare} = \text{Speed} \times \text{Time} = 6.67 \text{ m/s} \times 120 \text{ s} = 800 \text{ m} \]

### Step 3: Calculate the total distance between the dog and the hare when the dog starts
Initially, the hare is 200 m away from the dog. After 2 minutes, the hare has covered 800 m, so the total distance between the dog and the hare when the dog starts chasing is:
\[ \text{Total distance} = 200 \text{ m} + 800 \text{ m} = 1000 \text{ m} \]

### Step 4: Convert the dog's speed from km/hr to m/s
The dog's speed is given as 32 km/hr. Converting this to m/s:
\[ \text{Speed of dog} = 32 \text{ km/hr} \times \frac{1}{3.6} \text{ m/s} = \frac{32}{3.6} \text{ m/s} \approx 8.89 \text{ m/s} \]

### Step 5: Calculate the relative speed of the dog with respect to the hare
Since both are moving in the same direction, the relative speed is:
\[ \text{Relative speed} = \text{Speed of dog} - \text{Speed of hare} = 8.89 \text{ m/s} - 6.67 \text{ m/s} = 2.22 \text{ m/s} \]

### Step 6: Calculate the time taken for the dog to overtake the hare
Using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} \]
\[ \text{Time} = \frac{1000 \text{ m}}{2.22 \text{ m/s}} \approx 450 \text{ seconds} \approx 7.5 \text{ minutes} \]

### Step 7: Calculate the distance from the starting point of the hare when overtaken
Now we need to find out how far the hare has run in that time:
\[ \text{Distance covered by hare} = \text{Speed of hare} \times \text{Time} \]
\[ \text{Distance} = 6.67 \text{ m/s} \times 450 \text{ s} \approx 3000 \text{ m} \]

### Final Results
- The dog overtakes the hare in approximately 7.5 minutes.
- The distance from the spot where the hare took flight when overtaken is approximately 3000 m.