A cat runs and jumps from one roof top to another that is 5 meters away and 3 meters below.  What is the minimum horizontal speed with which the cat must jump off the first roof in order to make it to the other?  *Round to the nearest whole number.*

To solve this problem, we need to analyze the cat's motion in two directions: horizontal and vertical.

### Vertical Motion:
The vertical displacement (\( \Delta y \)) is 3 meters below the starting point, so:
\[
\Delta y = -3 \, \text{m}.
\]

We can use the kinematic equation for vertical motion:
\[
\Delta y = v_{y0} t + \frac{1}{2} g t^2,
\]
where:
- \( v_{y0} = 0 \, \text{m/s} \) (since the cat is not initially moving vertically),
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity),
- \( t \) is the time of flight.

Substituting the known values:
\[
-3 = 0 + \frac{1}{2} (9.8) t^2,
\]
\[
t^2 = \frac{-6}{-9.8} = 0.6122,
\]
\[
t = \sqrt{0.6122} \approx 0.78 \, \text{seconds}.
\]

### Horizontal Motion:
Now, the horizontal displacement (\( \Delta x \)) is 5 meters, and horizontal speed (\( v_x \)) is constant since there's no acceleration in the horizontal direction.

The relationship between horizontal displacement and time is:
\[
\Delta x = v_x t.
\]
Solving for \( v_x \):
\[
v_x = \frac{\Delta x}{t} = \frac{5}{0.78} \approx 6.41 \, \text{m/s}.
\]

Rounding to the nearest whole number, the minimum horizontal speed the cat must have is:
\[
v_x \approx 6 \, \text{m/s}.
\]

### Answer:
The minimum horizontal speed is **6 m/s**.

Would you like further details or have any questions? 

Here are 5 related questions:
1. How does the height difference affect the required speed?
2. What happens if the height difference were 0 meters?
3. What is the effect of increasing the horizontal distance?
4. How does air resistance affect real-world jumps?
5. How would the time change if the gap was 10 meters instead of 5?

**Tip:** For projectile motion, vertical and horizontal components of motion are independent of each other.

A cat runs and jumps from one rooftop to another that is 5 meters away and 3 meters below. What is the minimum horizontal speed with which the cat must jump off the first roof in order to make it to the other? Round to the nearest whole number.

Discover how to calculate the minimum horizontal speed a cat must have to jump 5 meters across to a lower rooftop 3 meters below. This problem involves kinematics, using key projectile motion equations. Suitable for students in Grades 9-12.

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