We can solve this by breaking the problem into two parts:

1. Velocity of the boat with respect to the ground:

• Boat velocity with respect to water = $12 \, \text{km/h}$ (upstream)
• Water velocity with respect to ground = $8 \, \text{km/h}$ (downstream)

Since the boat is moving upstream (against the current), we subtract the water's velocity from the boat's velocity.

$V_{\text{boat/ground}} = V_{\text{boat/water}} - V_{\text{water/ground}} = 12 \, \text{km/h} - 8 \, \text{km/h} = 4 \, \text{km/h}$

So, the boat's velocity with respect to the ground is 4 km/h upstream.

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2. Velocity of the child with respect to the ground:

• Child's velocity with respect to boat = $6 \, \text{km/h}$ (towards the back, which is opposite to the boat's direction)
• Boat's velocity with respect to ground = $4 \, \text{km/h}$ (upstream)

The child is walking toward the back of the boat, so we subtract the child's velocity from the boat's velocity.

$V_{\text{child/ground}} = V_{\text{boat/ground}} - V_{\text{child/boat}} = 4 \, \text{km/h} - 6 \, \text{km/h} = -2 \, \text{km/h}$

This negative result means the child is moving downstream with respect to the ground.

So, the child's velocity with respect to the ground is 2 km/h downstream.

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Summary:

1. Boat's velocity with respect to the ground: $4 \, \text{km/h}$ upstream.
2. Child's velocity with respect to the ground: $2 \, \text{km/h}$ downstream.

Would you like further clarification or details on any part of this solution?

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Here are 5 related questions to explore further:

1. What would happen to the boat's velocity if the water was moving faster than the boat?
2. How would the child's velocity change if they walked in the same direction as the boat?
3. What if the boat moved downstream instead of upstream—how would that affect the calculations?
4. How does the reference frame affect the perceived velocity of objects?
5. Can you calculate the relative velocity of two boats moving in opposite directions?

Tip: Always make sure to identify the reference frame when solving velocity problems in physics.