To solve this problem, we use kinematic equations for projectile motion. We'll break the initial velocity ($v_0$) into horizontal and vertical components, analyze the vertical motion for maximum height and time of flight, and then use this to answer the given questions.

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Given:

• Launch angle ($\theta$) = $30^\circ$
• Maximum height ($h_{\text{max}}$) = $0.6 \, \text{m}$
• Acceleration due to gravity ($g$) = $9.8 \, \text{m/s}^2$

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Step 1: Find the vertical velocity ($v_{y0}$) at launch

At the maximum height, the vertical velocity becomes zero. Using the kinematic equation: $v_y^2 = v_{y0}^2 - 2 g h_{\text{max}}$ At maximum height, $v_y = 0$. Thus: $0 = v_{y0}^2 - 2 g h_{\text{max}}$ $v_{y0} = \sqrt{2 g h_{\text{max}}}$ Substitute the known values: $v_{y0} = \sqrt{2 \cdot 9.8 \cdot 0.6} = \sqrt{11.76} \approx 3.43 \, \text{m/s}$

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Step 2: Find the initial velocity ($v_0$) at launch

The vertical component of velocity is related to the total initial velocity by: $v_{y0} = v_0 \sin\theta$ Rearrange to solve for $v_0$: $v_0 = \frac{v_{y0}}{\sin\theta}$ Substitute $v_{y0} = 3.43 \, \text{m/s}$ and $\sin(30^\circ) = 0.5$: $v_0 = \frac{3.43}{0.5} \approx 6.86 \, \text{m/s}$

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Step 3: Find the total time of flight ($T$)

The total time of flight is the time it takes for the frog to go up and come down. The time to reach maximum height ($t_{\text{up}}$) is given by: $t_{\text{up}} = \frac{v_{y0}}{g}$ Substitute $v_{y0} = 3.43 \, \text{m/s}$ and $g = 9.8 \, \text{m/s}^2$: $t_{\text{up}} = \frac{3.43}{9.8} \approx 0.35 \, \text{s}$ The total time of flight is twice this value: $T = 2 t_{\text{up}} = 2 \cdot 0.35 \approx 0.70 \, \text{s}$

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Final Answers:

(a) The velocity with which the frog leaves the ground is approximately: $\boxed{6.86 \, \text{m/s}}$ (b) The total time the frog is in the air is approximately: $\boxed{0.70 \, \text{s}}$

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Would you like detailed step-by-step derivations for any part?

Here are 5 related questions for deeper understanding:

1. How can you calculate the range of the frog's jump?
2. How does the launch angle affect the maximum height?
3. What happens to the time of flight if the frog jumps with a steeper angle but the same speed?
4. How would the calculations change if the ground was not level (e.g., landing on a hill)?
5. Can you calculate the horizontal distance covered by the frog?

Tip: Always separate the motion into horizontal and vertical components when solving projectile problems!