To calculate the vertical speed $v_0$ of the kangaroo when it leaves the ground, we can use the kinematic equation that relates the maximum height $h$ reached to the initial vertical speed $v_0$, under the influence of gravity $g$:

$v_0^2 = 2gh$

Where:

• $h = 2.2 \, \text{m}$ is the maximum height.
• $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity.

Solving for $v_0$:

$v_0 = \sqrt{2gh}$

Substituting the values:

$v_0 = \sqrt{2 \times 9.8 \, \text{m/s}^2 \times 2.2 \, \text{m}} = \sqrt{43.12} \, \text{m}^2/\text{s}^2 \approx 6.57 \, \text{m/s}$

So, the vertical speed when the kangaroo leaves the ground is approximately 6.57 m/s.

Would you like more details or have any questions? Here are some related questions:

1. How would the vertical speed change if the object height increased to 3 meters?
2. What would happen to the vertical speed if the gravity was different, like on the Moon?
3. Can we determine the time the kangaroo spends in the air with this information?
4. How does the initial vertical speed relate to the range of the kangaroo's jump?
5. What other factors, besides gravity, could affect the kangaroo's vertical speed?

Tip: The square root function used in this equation indicates that the vertical speed is sensitive to changes in height, but the relationship is not linear.