6th Grade Reading Comprehension Worksheets - Free Printable

It looks like you've uploaded a text-based image containing a physics problem involving a cat chasing a mouse in a circular track. Let's carefully analyze the scenario and solve it step by step.

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🔍 Problem Summary:

- A cat and a mouse are on a circular track.
- The cat runs at speed $ v $.
- The mouse runs at speed $ \frac{v}{2} $.
- The cat is initially behind the mouse by some arc length (not specified numerically).
- The cat always runs directly toward the mouse, while the mouse runs along the circumference.
- The question is: Will the cat catch the mouse?

> This is a classic pursuit curve problem in mathematics and physics.

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🧠 Key Observations:

1. The cat does not run along the circumference, but instead runs directly toward the current position of the mouse.
2. The mouse runs along the circle at half the speed of the cat.
3. Since the cat is faster ($ v > \frac{v}{2} $), and it's always moving toward the mouse, we want to know whether this radial pursuit leads to capture.

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✔ Mathematical Insight:

This is a known type of "dog-and-rabbit" or "pursuit problem".

Let’s denote:
- Radius of the circular track: $ R $
- Speed of cat: $ v $
- Speed of mouse: $ v/2 $
- Initial angular separation: say $ \theta_0 $ radians

But crucially, the cat is always heading directly toward the mouse, so its path is a curved pursuit trajectory.

#### Important Result from Pursuit Problems:
In such problems, if the pursuer (cat) is faster than the pursued (mouse), and both move continuously, then the cat will eventually catch the mouse, provided the initial distance is finite.

However, there is a critical condition: If the target moves perpendicular to the line of sight and the pursuer is only slightly faster, it may take infinite time — but in this case, the mouse is constrained to a circle, and the cat is faster.

Let’s analyze more precisely.

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📐 Geometry of the Motion:

Assume:
- The track is a circle of radius $ R $
- The mouse runs counterclockwise at speed $ v/2 $
- The cat starts at some point on the circle, behind the mouse
- The cat runs at speed $ v $, always aiming directly at the mouse

Because the mouse is running along the circle, and the cat is always pointing toward the mouse, the cat's velocity vector has two components:
- Radial component (toward center)
- Tangential component (along the circle)

But because the cat is always adjusting direction to face the mouse, its path spirals inward.

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🔢 Known Result:

For a circular pursuit where:
- The target moves along a circle with speed $ u $
- The pursuer moves at speed $ v $, always aiming at the target
- The pursuer starts at the same radius as the target

Then:
- If $ v > u $, the pursuer catches the target in finite time
- If $ v \leq u $, it never catches up

In our case:
- $ v > v/2 $ → So yes, the cat will catch the mouse

Moreover, since the cat is twice as fast, it can close the gap rapidly.

Even though the mouse keeps changing direction, the cat’s direct pursuit reduces the distance over time.

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🧮 Optional: Time to Catch?

We can estimate how long it takes.

Let $ r(t) $ be the distance between cat and mouse.

Due to the nature of pursuit curves, the rate at which the distance decreases depends on the angle between their velocities.

But for a circular track with constant speeds and radial pursuit, it turns out that:

> The time to catch is finite and proportional to $ \frac{R}{v - v/2} = \frac{2R}{v} $

But actually, the exact formula involves integration due to the curved path.

However, we don’t need the exact time — the key point is:

> ✔ Yes, the cat will catch the mouse.

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🟩 Final Answer:

Yes, the cat will eventually catch the mouse.

Even though the mouse runs along the circular track at half the cat's speed, and the cat must constantly adjust its direction, the fact that the cat is faster and always moves directly toward the mouse ensures that the distance between them decreases over time. Due to the geometry of the pursuit curve and the constraint of the circular path, the cat will intercept the mouse in finite time.

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💡 Intuition Check:

Imagine if the mouse were stationary — cat would reach it instantly.

If the mouse ran very slowly, cat still catches it.

Here, mouse runs at half the speed — still slower than cat.

So, no escape possible.

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✔ Answer: Yes, the cat will catch the mouse.