Question

Part 1: A cheetah can run at a maximum speed 101 km/h and a gazelle can run at a maximum speed of 78.4 km/h. If both animals are running at full speed, with the gazelle 82.7 m ahead, how long before the cheetah catches its prey? Answer in units of s.

Part 2: 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.

Answer 1

To solve the problem, we'll start with Part 1.

### Part 1: Time until the cheetah catches the gazelle

1. **Convert speeds from km/h to m/s:**
\[
\text{Speed of cheetah} = 101 \text{ km/h} = \frac{101 \times 1000}{3600} \approx 28.06 \text{ m/s}
\]
\[
\text{Speed of gazelle} = 78.4 \text{ km/h} = \frac{78.4 \times 1000}{3600} \approx 21.78 \text{ m/s}
\]

2. **Determine the relative speed of the cheetah compared to the gazelle:**
\[
\text{Relative speed} = \text{Speed of cheetah} - \text{Speed of gazelle} = 28.06 \text{ m/s} - 21.78 \text{ m/s} = 6.28 \text{ m/s}
\]

3. **Calculate the time it takes for the cheetah to close the distance:**
The cheetah is 82.7 m behind the gazelle, so we can use the formula:
\[
\text{Time} = \frac{\text{Distance}}{\text{Relative speed}} = \frac{82.7 \text{ m}}{6.28 \text{ m/s}} \approx 13.15 \text{ s}
\]

Thus, the cheetah will catch the gazelle in approximately **13.15 seconds**.

### Part 2: Minimum distance for the gazelle to escape

1. **Calculate the distance the cheetah can run in 7.5 seconds:**
\[
\text{Distance cheetah can run} = \text{Speed of cheetah} \times \text{Time} = 28.06 \text{ m/s} \times 7.5 \text{ s} \approx 210.45 \text{ m}
\]

2. **Calculate the distance the gazelle can run in the same 7.5 seconds:**
\[
\text{Distance gazelle can run} = \text{Speed of gazelle} \times \text{Time} = 21.78 \text{ m/s} \times 7.5 \text{ s} \approx 163.35 \text{ m}
\]

3. **Find the minimum distance the gazelle must be ahead:**
The gazelle needs to have a minimum distance that ensures it is ahead after both have run for 7.5 seconds. The minimum distance can be calculated as the difference in distances they cover in 7.5 seconds:
\[
\text{Minimum lead} = \text{Distance cheetah can run} - \text{Distance gazelle can run} = 210.45 \text{ m} - 163.35 \text{ m} \approx 47.1 \text{ m}
\]

Thus, to ensure a chance of escape, the gazelle must be at least **47.1 meters** ahead of the cheetah.