To solve this problem, we need to analyze the relative motion of the dog and the rabbit.

Step 1: Convert time taken for each animal to complete a jump.

The time taken by the dog to make 3 jumps is equal to the time taken by the rabbit to make 4 jumps. This means that the ratio of their speeds is:

$\text{Speed of dog} = \frac{3}{4} \times \text{Speed of rabbit}$

Step 2: Calculate the distances traveled by each animal in one jump.

• The rabbit covers $1.75 \, \text{m}$ in one jump.
• The dog covers $2.75 \, \text{m}$ in one jump.

Step 3: Determine the speeds of the animals.

Let the time taken for the rabbit to make one jump be $t_r$, and the time taken for the dog to make one jump be $t_d$.

• Speed of the rabbit = $\frac{1.75 \, \text{m}}{t_r}$
• Speed of the dog = $\frac{2.75 \, \text{m}}{t_d}$

From the problem, we know that: $t_d = \frac{4}{3} t_r$ So, the dog’s speed relative to the rabbit will be: $\text{Speed of dog relative to rabbit} = \frac{2.75}{\frac{4}{3}} = \frac{2.75 \times 3}{4} = 2.0625 \, \text{m per unit time}$ The rabbit’s speed is: $\text{Speed of rabbit} = \frac{1.75}{t_r}$

Step 4: Calculate the relative distance.

The rabbit is initially ahead by $125 \times 1.75 = 218.75 \, \text{m}$.

Step 5: Determine how many jumps it takes for the dog to catch the rabbit.

To catch the rabbit, the dog has to close the gap of $218.75 \, \text{m}$ at a rate of: $\text{Relative speed} = 2.0625 - 1.75 = 0.3125 \, \text{m per unit time}$

The time required for the dog to close the gap is: $\text{Time to catch the rabbit} = \frac{218.75}{0.3125} = 700 \, \text{time units}$

Step 6: Convert time units into jumps.

In the time $t_d$ (the time it takes for the dog to make one jump), the dog will complete: $\text{Number of jumps} = \frac{700}{t_d}$

Since $t_d = \frac{4}{3} t_r$, we can express the number of jumps for the dog as: $\text{Number of jumps} = \frac{700}{\frac{4}{3} t_r} = \frac{700 \times 3}{4 t_r} = 525 \, \text{jumps}$

So, the dog will catch the rabbit after 525 jumps.

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Would you like further clarification on any part of this problem? Here are some additional questions for deeper understanding:

1. How does the relative speed of two moving objects affect the time it takes for one object to catch the other?
2. How can you use unit conversion in physics problems involving speed and time?
3. How do you calculate the number of jumps or steps required to close a gap between two moving objects?
4. How does the ratio of time taken to complete a task (e.g., jumping) influence the overall speed comparison?
5. What role does the initial distance between two objects play in solving relative motion problems?

Tip: Always identify the reference frame you're using when solving relative motion problems—whether it's for one object or comparing two moving objects.