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Chapter 6: Problem 59

A 67.0-kg person jumps from rest off a 3.00-m-high tower straight down into the water. Neglect air resistance. She comes to rest 1.10 m under the surface of the water. Determine the magnitude of the average force that the water exerts on the diver. This force is non conservative.

Short Answer

Expert verified
The average force exerted by the water is approximately 1792.58 N.

Step by step solution

01

Calculate the Diver's Initial Velocity

The diver jumps from a height of 3.00 m. We can use the kinematic equation to find her velocity just before entering the water: \( v^2 = u^2 + 2gh \) where \( u = 0 \) (initial velocity), \( g = 9.8\, \text{m/s}^2 \) (acceleration due to gravity), and \( h = 3.00\, \text{m} \). The velocity \( v \) when she hits the water is calculated as follows:\[ v = \sqrt{0 + 2(9.8)(3.00)} \]\[ v = \sqrt{58.8} \approx 7.67\, \text{m/s} \]
02

Calculate the Diver's Change in Momentum

Once in the water, the diver comes to rest, indicating her final velocity is \( 0 \). Her initial velocity when she enters the water is \( 7.67\, \text{m/s} \). Using her mass, \( m = 67.0\, \text{kg} \), the change in momentum \( \Delta p \) is:\[ \Delta p = m \times (v_f - v_i) = 67.0 \times (0 - 7.67) \approx -514.89\, \text{kg} \cdot \text{m/s} \]
03

Calculate the Deceleration Under Water

The diver comes to rest after traveling 1.10 m in the water. Using the equation \( v_f^2 = v_i^2 + 2as \) and solving for \( a \) where \( v_f = 0 \), \( v_i = 7.67 \), and \( s = 1.10 \), we have:\[ 0 = (7.67)^2 + 2a(1.10) \]\[ a = -\frac{(7.67)^2}{2 \times 1.10} \approx -26.74\, \text{m/s}^2 \]
04

Determine the Average Force Exerted by the Water

The average force exerted by the water can be found using Newton's second law: \( F = m \times a \). With \( m = 67.0\, \text{kg} \) and \( a = -26.74 \), the force \( F \) is:\[ F = 67.0 \times 26.74 \approx 1792.58\, \text{N} \]Note that the force is positive as it is exerted upwards against the direction of motion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

kinematic equations
Kinematic equations are a set of equations that describe the motion of objects under constant acceleration. They help predict future motion based on initial conditions and forces. In our exercise, the diver begins at rest before jumping off a 3.00-meter tower. By using the kinematic equation, \( v^2 = u^2 + 2gh \), we can determine her velocity just before she hits the water.
This equation works because it accounts for the acceleration due to gravity \( (g = 9.8 \text{ m/s}^2) \) and the initial velocity \( (u = 0 \text{ m/s}) \). Applying the numbers, the diver's velocity just before impact with water comes to approximately 7.67 m/s.
momentum change
Momentum is a product of an object's mass and velocity. Change in momentum occurs when an object's speed changes, which usually happens when a force is applied. For the diver, her momentum changes as she goes from moving to resting in the water.
Initially, her momentum is non-zero due to her speed upon striking the water. With an initial velocity of 7.67 m/s and mass of 67.0 kg, her change in momentum becomes \(-514.89\, \text{kg} \cdot \text{m/s}\). This reduction in momentum indicates a considerable force exerted by the water to stop the diver.
deceleration
Deceleration is the reduction of speed or velocity. In this problem, it's the process of the diver slowing down upon entering the water. It is essentially negative acceleration and is determined by using the kinematic equation again.
Here, we know her initial velocity (7.67 m/s), final velocity (0 m/s after stopping), and the distance over which she decelerates (1.10 m). By substituting these values into the equation \( v_f^2 = v_i^2 + 2as \), we can calculate the deceleration as approximately -26.74 m/s虏. The negative sign indicates a reduction in speed in the opposite direction of her entering velocity.
Newton's second law
Newton's second law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration (\( F = ma \)). It's a key principle in understanding how forces result in changes in motion.
In the context of the diver, once we know the mass (67.0 kg) and the deceleration (-26.74 m/s虏), we can calculate the force the water applies to stop her. This calculation reveals an average force of approximately 1792.58 N. Importantly, this force is directed upwards as the water resists her downwards motion, allowing her to come to a complete stop.

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Most popular questions from this chapter

In the sport of skeleton a participant jumps onto a sled (known as a skeleton) and proceeds to slide down an icy track, belly down an head first. In the 2010 Winter Olympics, the track had sixteen turns and dropped 126 m in elevation from top to bottom. (a) In the absence of non conservative forces, such as friction and air resistance, what would be the speed of a rider at the bottom of the track? Assume that the speed at the beginning of the run is relatively small and can be ignored. (b) In reality, the gold-medal winner (Canadian Jon Montgomery) reached the bottom in one heat with a speed of 40.5 m/s (about 91 mi/h). How much work was done on him and his sled (assuming a total mass of 118 kg) by non conservative forces during this heat?

You are working out on a rowing machine. Each time you pull the rowing bar (which simulates the oars) toward you, it moves a distance of 1.2 m in a time of 1.5 s. The readout on the display indicates that the average power you are producing is 82 W. What is the magnitude of the force that you exert on the handle?

The motor of a ski boat generates an average power of \(7.50 \times 10^{4} \mathrm{W}\) when the boat is moving at a constant speed of 12 \(\mathrm{m} / \mathrm{s}\) . When the boat is pulling a skier at the same speed, the engine must generate an average power of \(8.30 \times 10^{4} \mathrm{W}\) . What is the tension in the tow rope that is pulling the skier?

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