Question

The great, gray-green, greasy Zambezi River flows over Victoria Falls in south central Africa. The falls are approximately \(108 \mathrm{~m}\) high. If the river is flowing horizontally at \(3.60 \mathrm{~m} / \mathrm{s}\) just before going over the falls, what is the speed of the water when it hits the bottom? Assume that the water is in free fall as it drops.

Short answer

The speed of the water at the bottom is approximately 46.2 m/s.

Step-by-step solution

Understand the Problem

We need to find the final speed of the water when it hits the bottom of the falls using the given height of the falls and the initial horizontal speed of the water.

Break Down the Motion

The motion of the water can be separated into horizontal and vertical components. The horizontal speed remains constant at \(3.60 \ \text{m/s}\), while the vertical speed changes due to gravity.

Calculate Vertical Velocity

Use the equation for final velocity under free fall: \( v_y = \sqrt{2gh}\), where \(g = 9.8 \ \text{m/s}^2\) is the acceleration due to gravity and \(h = 108 \ \text{m}\) is the height of the falls. Substitute the values to get \(v_y = \sqrt{2 \times 9.8 \times 108} \approx 46.1 \ \text{m/s}\).

Determine Overall Speed

The final speed of the water can be calculated using the Pythagorean theorem since the horizontal and vertical motions are perpendicular: \(v = \sqrt{v_x^2 + v_y^2}\), where \(v_x = 3.60 \ \text{m/s}\) and \(v_y = 46.1 \ \text{m/s}\). Calculate to get \(v = \sqrt{(3.60)^2 + (46.1)^2} \approx 46.2 \ \text{m/s}\).

Key concepts

Free Fall

Free fall describes the motion of an object dropped under the influence of gravity alone. In this scenario, the water from the Zambezi River, after going over the edge of Victoria Falls, is affected only by gravitational force. Thus, it experiences free fall. Gravity accelerates the water vertically downward at a constant rate of approximately 9.8 m/s². This acceleration is uniform, meaning the velocity of the water increases consistently until it hits the bottom of the falls.
The vertical component of the water's velocity can be calculated using the equation for free fall: \[ v_y = \sqrt{2gh} \]

• Where \(v_y\) is the final vertical velocity.
• \(g\) is the acceleration due to gravity.
• \(h\) is the height of the fall.

By substituting the known values, we find that the vertical velocity of the water is approximately 46.1 m/s, illustrating the significant impact of gravitational force during the river's descent.

Kinematics

Kinematics is the branch of physics focusing on the motion of objects. It does not consider the forces causing the motion. For Victoria Falls, the motion has two primary components: horizontal and vertical.

In horizontal motion, the water retains its original speed of 3.60 m/s, unaffected by gravity. This happens because there's no horizontal acceleration impacting it directly. Meanwhile, the vertical motion is influenced by gravity, causing the water to accelerate as it falls.

The crucial point in understanding kinematics here is recognizing that these two components are independent of each other. Even as the water accelerates vertically, its horizontal velocity remains constant. This separation of motion simplifies the problem, allowing us to handle each component distinctly and later combine them to find the overall motion.

Velocity Calculation

Velocity encompasses both the speed and direction of a moving object. Here, calculating the water's velocity as it hits the base of Victoria Falls involves considering both its horizontal and vertical components.
By applying the Pythagorean theorem, we can combine these perpendicular components to determine the resultant velocity:\[v = \sqrt{v_x^2 + v_y^2}\]

• \(v_x = 3.60 \, \text{m/s}\)
• \(v_y = 46.1 \, \text{m/s}\)

After calculation, the water's speed upon impact is approximately 46.2 m/s. This indicates how both the initial horizontal motion and the acceleration due to gravity interact to form the final velocity. Understanding this concept helps students grasp how different vectors of motion combine to dictate an object's path.