To calculate the final velocity of the pebble just before it hits the ground, we can use the following kinematic equation:

$v^2 = u^2 + 2as$

Where:

• $v$ is the final velocity (what we need to find),
• $u = -6.70 \, \text{m/s}$ is the initial velocity (negative because it is downward),
• $a = -9.8 \, \text{m/s}^2$ is the acceleration due to gravity (negative because it acts downward),
• $s = -38.8 \, \text{m}$ is the displacement (negative because the pebble moves downward from the bridge to the ground).

Step 1: Substitute the values into the equation:

$v^2 = (-6.70)^2 + 2(-9.8)(-38.8)$

Step 2: Perform calculations step by step:

1. $(-6.70)^2 = 44.89$,
2. $2(-9.8)(-38.8) = 759.04$,
3. Add these together: $v^2 = 44.89 + 759.04 = 803.93$.

Step 3: Solve for $v$:

$v = -\sqrt{803.93} \quad (\text{taking the negative root, since the velocity is downward}).$ $v = -28.36 \, \text{m/s}.$

Final Answer:

The final velocity of the pebble just before it hits the ground is: $\boxed{-28.36 \, \text{m/s}}$

Would you like a detailed breakdown of the steps or clarifications?

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Here are 5 related questions to deepen your understanding:

1. What is the total time it takes for the pebble to hit the ground?
2. How would the final velocity change if the initial velocity were upwards?
3. What happens to the displacement value if the pebble is thrown upwards initially?
4. How can energy conservation principles verify the final velocity?
5. How does the acceleration due to gravity affect objects in free fall?

Tip: Always pay attention to the signs of vectors in kinematic equations—they determine direction!