The cheetah can maintain its maximum speed for only 7.5 s. What is the minimum distance the gazelle must be ahead of the cheetah to have a chance of escape? (After 7.5 s the speed of cheetah is less than that of the gazelle.) Answer in units of m.

To solve this problem, we need to compare the motion of the cheetah and the gazelle over the critical time of 7.5 seconds. 

Assume:
- The cheetah's maximum speed is \( v_{\text{cheetah}} \), and it can maintain this speed for 7.5 seconds.
- The gazelle's constant speed is \( v_{\text{gazelle}} \), and it maintains this speed throughout.
- The minimum distance the gazelle must be ahead for it to escape is \( d \).

We know that after 7.5 seconds, the cheetah can no longer maintain its speed and will slow down. So, during the 7.5 seconds, the cheetah needs to cover a distance to reach the gazelle, meaning the distance the gazelle needs to be ahead of the cheetah is the difference between the distances traveled by both animals in that time.

### Step-by-Step Solution:

1. The distance covered by the cheetah during 7.5 seconds is:
   \[
   d_{\text{cheetah}} = v_{\text{cheetah}} \times 7.5
   \]

2. The distance covered by the gazelle during the same time is:
   \[
   d_{\text{gazelle}} = v_{\text{gazelle}} \times 7.5
   \]

3. For the gazelle to escape, the initial distance \( d \) must be at least equal to the distance the cheetah would cover in 7.5 seconds minus the distance the gazelle covers:
   \[
   d = d_{\text{cheetah}} - d_{\text{gazelle}} = (v_{\text{cheetah}} - v_{\text{gazelle}}) \times 7.5
   \]

So, the minimum distance the gazelle must be ahead is:

\[
d = (v_{\text{cheetah}} - v_{\text{gazelle}}) \times 7.5
\]

### Required Information:
- \( v_{\text{cheetah}} \) (the cheetah's maximum speed)
- \( v_{\text{gazelle}} \) (the gazelle's constant speed)

If you provide the speeds of the cheetah and gazelle, I can calculate the exact value of the minimum distance.

Would you like to provide those values or have further clarification? 

Here are 5 related questions for further exploration:

1. How do acceleration and deceleration affect pursuit dynamics in predators and prey?
2. What is the biological significance of a cheetah's short sprinting time?
3. How can one calculate the effect of terrain on animal pursuit speeds?
4. What other animals have a similar speed profile to the cheetah?
5. How does the energy expenditure of fast animals change with time?

**Tip:** In motion problems, always consider the total time over which speeds are constant versus changing.

Discover how to solve a kinematics problem involving a cheetah and a gazelle. Learn how to calculate the minimum distance the gazelle must be ahead of the cheetah for a chance of escape, using key formulas and concepts. Suitable for high school students in Grades 9-12.

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