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When considering the velocity of a boat, we need to think about how fast the boat moves in relation to the water around it. This is known as the boat's velocity relative to the water. In the given problem, the boat travels upstream at 14 km/h against the water's current, which moves at 9 km/h. To determine how quickly and in which direction the boat moves relative to the ground, it's essential to account for the effect of the water's current.

When the boat travels upstream, it is moving opposite to the direction of the water current. Thus, to find the boat's velocity concerning the ground, we subtract the velocity of the water from the velocity of the boat itself: \( v_{bg} = v_{bw} - v_{w} \).

In this instance, it simplifies to \(14 \text{ km/h} - 9 \text{ km/h} = 5 \text{ km/h} \). This means the boat moves at 5 km/h relative to the ground, upstream, or in the positive direction of the x-axis.

Understanding reference frames is crucial when dealing with velocities. A reference frame is essentially a point of view from which we measure and observe motion. For this problem, three different reference frames are considered:

• The water as the reference frame for the boat. Here, the boat's velocity is 14 km/h.
• The ground as the reference frame for both the water and the boat.
• The boat itself as the reference frame for the child. The child is moving at 6 km/h within this frame.

By switching reference frames, we can determine velocities from different perspectives. When we compute the velocity of the boat with respect to the ground, we are switching from the water's frame to the ground's frame. This switch helps us see how the boat actually moves over solid ground, factoring in the opposing current from the river.

Motion in one dimension simplifies problems since movement occurs along a single straight line. In this context, both the boat's and the child's movements occur on the x-axis—marking it as one-dimensional motion.

Understanding this setup is crucial because it limits velocity to two possible directions: positive or negative. In the exercise, the boat moves upstream (positive direction), and the child walking in the opposite direction brings about a negative velocity component with respect to the boat.

The mathematics of motion in one dimension revolves around simple addition or subtraction of velocities, such as when calculating how the child's walking speed (opposing the boat's direction) influences their total velocity relative to ground. Thus, this form of motion simplifies the complexity often present in multi-dimensional motion problems.

Vector addition is a method used to combine different velocities when they are in effect simultaneously. In our example, the boat's velocity with respect to the ground and the child's velocity demonstrate this concept.

Since the child walks in the opposite direction of the boat's movement, vector addition involves subtracting the child's walking speed from the boat's velocity relative to the ground. This concept applies as follows: \( v_{cg} = v_{bg} - v_{cb} \), where \(v_{cg}\) is the child's velocity with respect to the ground, \(v_{bg}\) is the boat's velocity relative to the ground, and \(v_{cb}\) is the child's velocity relative to the boat.

The result \(-1 \text{ km/h} \) denotes that the child's net velocity is directed downstream compared to the ground, highlighting the countereffect of vector addition when directions oppose each other.