Problem 70 - Fundamentals of Physics

Question

A boat is traveling upstream in the positive direction of an \(x\) axis at \(14 \mathrm{~km} / \mathrm{h}\) with respect to the water of a river. The water is flowing at \(9.0 \mathrm{~km} / \mathrm{h}\) with respect to the ground. What are the (a) magnitude and (b) direction of the boat's velocity with respect to the ground? A child on the boat walks from front to rear at \(6.0 \mathrm{~km} / \mathrm{h}\) with respect to the boat. What are the (c) magnitude and (d) direction of the child's velocity with respect to the ground?

Step-by-Step Solution

Answer

(a) 5.0 km/h upstream, (c) 1.0 km/h downstream.

Step 1: Analyze the Boat's Velocity Relative to the Ground

The boat is moving upstream at \(14 \, \text{km/h}\) with respect to the water and the water itself is flowing at \(9.0 \, \text{km/h}\) with respect to the ground. Since the boat is moving upstream against the current, the velocity of the boat with respect to the ground \(v_{bg}\) is given by the equation:\[ v_{bg} = v_{bw} - v_{wg} = 14 \, \text{km/h} - 9.0 \, \text{km/h} = 5.0 \, \text{km/h} \]

Step 2: Determine the Boat's Direction Relative to the Ground

The boat's velocity with respect to the ground is positive because it is moving upstream, which is defined as the positive direction for this problem. Therefore, the direction of the boat's velocity with respect to the ground is upstream.

Step 3: Analyze the Child's Velocity Relative to the Ground

The child is walking from the front to the rear of the boat at \(6.0 \, \text{km/h}\) with respect to the boat, which is moving upstream. Considering the boat's movement, the child's velocity \(v_{cg}\) with respect to the ground is given by:\[ v_{cg} = v_{cb} + v_{bg} = -6.0 \, \text{km/h} + 5.0 \, \text{km/h} = -1.0 \, \text{km/h} \] The negative sign indicates that the velocity direction is opposite to the positive direction.

Step 4: Determine the Child's Direction Relative to the Ground

The child's velocity with respect to the ground is \(-1.0 \, \text{km/h}\), indicating motion in the negative direction. This means the child is moving downstream with respect to the ground.

Key Concepts

Upstream and Downstream

When we discuss objects moving in water, like a boat, terms like upstream and downstream come into play. These terms help describe the relative direction of movement concerning the water's current.
If a boat moves upstream, it is going against the direction of water flow. Downstream means moving along with the current. For example, if a river flows north and a boat travels south, it is going upstream. Conversely, moving northward with the flow is traveling downstream.
Understanding these directions is crucial because they affect the calculations of velocity in relation to other objects, like when a boat moves in a current.

Velocity with Respect to Ground

When determining an object's velocity with respect to the ground, we need to consider both its speed and its direction. This concept is especially important when objects, like boats or people, are in a moving medium such as water or air.
For example, a boat moving upstream would have its velocity decreased by the current's speed. This is because you subtract the current's velocity from the boat's velocity. Conversely, when moving downstream, you add the current's velocity to the boat's velocity, resulting in a faster speed with respect to the ground.

• To find an object's velocity with respect to the ground, consider the combined effect of its movement and the flow of its surrounding medium.
• Use the formula for relative velocity: \[ v_{bg} = v_{bw} - v_{wg} \] for upstream and \[ v_{bg} = v_{bw} + v_{wg} \] for downstream.

Vector Addition

Vector addition is a mathematical operation that allows us to combine different velocities or forces. It’s crucial for solving problems involving relative velocities.
Imagine you’re facing both upstream and downstream movements, like with a boat in a river. You need to combine these different components to determine the actual velocity in relation to a fixed point, like the ground.

• Vectors are quantities that have both magnitude and direction, like velocity.
• Adding vectors involves considering both of these components: \[ v_{cg} = v_{cb} + v_{bg} \]for calculating the child's velocity with respect to the ground. Here, each part of the vector equation considers the different relative motions.

This calculation helps provide clarity on complex movements by breaking them down into simpler, additive parts.