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Chapter 6: Problem 53

At a carnival, you can try to ring a bell by striking a target with a \(9.00\) -kg hammer. In response, a \(0.400-\mathrm{kg}\) metal piece is sent upward toward the bell, which is \(5.00 \mathrm{~m}\) above. Suppose that \(25.0 \%\) of the hammer's kinetic energy is used to do the work of sending the metal piece upward. How fast must the hammer be moving when it strikes the target so that the bell just barely rings?

Short Answer

Expert verified
The hammer must be moving at approximately 4.17 m/s.

Step by step solution

01

Identify Given Values

The mass of the hammer is given as 9.00 kg. The mass of the metal piece is 0.400 kg. The height of the bell above the initial position of the metal piece is 5.00 m. The efficiency of energy transfer from the hammer to the metal piece is 25%. We need to find the speed of the hammer.
02

Determine the Energy Required to Reach the Bell

The potential energy required for the metal piece to reach the bell is given by the formula: \[ PE = mgh \] where \( m = 0.400 \) kg, \( g = 9.8 \) m/s虏, and \( h = 5.00 \) m. Calculate this value.
03

Calculate Potential Energy

Substitute the values into the formula:\[ PE = 0.400 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 5.00 \, \text{m} = 19.6 \, \text{J} \] This is the energy needed to elevate the metal piece to the height of the bell.
04

Determine Total Kinetic Energy Needed

Since only 25% of the hammer's kinetic energy is converted into the potential energy for the metal piece, calculate the total kinetic energy needed: \[ KE_{total} = \frac{19.6}{0.25} \] This will be the kinetic energy the hammer must initially have.
05

Calculate Total Initial Kinetic Energy

Perform the division:\[ KE_{total} = \frac{19.6}{0.25} = 78.4 \, \text{J} \] This is the initial kinetic energy the hammer needs to have to send the metal piece to the bell.
06

Find Hammer's Speed Using Kinetic Energy Formula

The kinetic energy of the hammer when it hits the target is given by:\[ KE = \frac{1}{2}mv^2 \] where \( m = 9.00 \) kg and \( KE = 78.4 \) J. Solve for the hammer's speed \( v \).
07

Solve for Speed

Rearrange the kinetic energy formula to solve for \( v \):\[ v^2 = \frac{2 \, KE}{m} = \frac{2 \, \times 78.4}{9.00} \]\[ v = \sqrt{\frac{156.8}{9}} \]Calculate \( v \).
08

Calculate Hammer鈥檚 Speed

Perform the calculation to find \( v \):\[ v = \sqrt{17.42} \approx 4.17 \, \text{m/s} \] This is the speed at which the hammer must strike the target for the bell to ring.

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Key Concepts

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

Kinetic Energy
Kinetic energy is a type of energy that an object possesses due to its motion. The faster an object moves, the more kinetic energy it has. It's given by the formula: \[ KE = \frac{1}{2}mv^2 \]where \( m \) represents the mass of the object and \( v \) is its velocity. For instance, in the carnival exercise, the hammer must possess a certain amount of kinetic energy to propel the metal piece upward. To determine this, we consider how much of the hammer's kinetic energy turns into potential energy to elevate the metal piece. Knowing only 25% is used, we calculate the total kinetic energy needed from the hammer, ensuring the metal can reach the bell's height.
Potential Energy
Potential energy refers to the energy held by an object because of its position relative to other objects. It鈥檚 often due to gravity, calculated with\[ PE = mgh \]where \( m \) is the mass, \( g \) the gravitational acceleration, and \( h \) the height above a reference point. In the problem, the metal piece needs a specific amount of potential energy to get to the bell. This amount is 19.6 J, as computed from the given height of 5 meters. This energy forms the crucial step for the metal piece to reach up and hit the bell, defying gravity due to its position above the launch point.
Energy Efficiency
Energy efficiency measures how well energy is transferred from one form to another. It's expressed as a percentage. In real-world applications, like the carnival game, only a portion of the hammer's kinetic energy effectively converts into potential energy of the rising metal piece. Here, that efficiency is 25%. This means most of the hammer鈥檚 energy is lost or not used in propelling the metal upward. Knowing this, we determine the initial required kinetic energy of the hammer to ensure enough potential energy is available to reach the bell height. This typical loss in efficiency illustrates why devices rarely have 100% energy conversion.
Kinematics
Kinematics is the branch of physics that describes the motion of objects, without deeply considering the forces that cause these movements. It helps in understanding and calculating how an object like the hammer moves to produce desired effects. In this context, we consider the motion needed for the hammer to strike a target and send the metal piece upward. This involves calculating the necessary speed using known energy principles. By understanding the speed required given specific kinetic energy, one can predict and control the outcomes of various physical motions effectively.

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Most popular questions from this chapter

A car accelerates uniformly from rest to \(20.0 \mathrm{~m} / \mathrm{s}\) in \(5.6 \mathrm{~s}\) along a level stretch of road. Ignoring friction, determine the average power required to accelerate the car if (a) the weight of the car is \(9.0 \times 10^{3} \mathrm{~N},\) and \((\mathrm{b})\) the weight of the car is \(1.4 \times 10^{4} \mathrm{~N}\)

Multiple-Concept Example 5 reviews many of the concepts that play a role in this problem. An extreme skier, starting from rest, coasts down a mountain slope that makes an angle of \(25.0^{\circ}\) with the horizontal. The coefficient of kinetic friction between her skis and the snow is 0.200 . She coasts down a distance of \(10.4 \mathrm{~m}\) before coming to the edge of a cliff. Without slowing down, she skis off the cliff and lands downhill at a point whose vertical distance is \(3.50 \mathrm{~m}\) below the edge. How fast is she going just before she lands?

A water-skier is being pulled by a tow rope attached to a boat. As the driver pushes the throttle forward, the skier accelerates. (a) What type of energy is changing? (b) Is the work being done by the net external force acting on the skier positive, zero, or negative? Why? (c) How is this work related to the change in the energy of the skier? A 70.3 -kg water-skier has an initial speed of \(6.10 \mathrm{~m} / \mathrm{s}\). Later, the speed increases to \(11.3 \mathrm{~m} / \mathrm{s}\). Determine the work done by the net external force acting on the skier.

Bicyclists in the Tour de France do enormous amounts of work during a race. For example, the average power per kilogram generated by Lance Armstrong \((m=75.0)\) is \(6.50 \mathrm{~W}\) per kilogram of his body mass. (a) How much work does he do during a \(135-\mathrm{km}\) race in which his average speed is \(12.0 \mathrm{~m} / \mathrm{s} ?\) (b) Often, the work done is expressed in nutritional Calories rather than in joules. Express the work done in part (a) in terms of nutritional Calories, noting that 1 joule \(=2.389 \times 10^{-4}\) nutritional Calories.

A 75.0 -kg skier rides a 2830 -m-long lift to the top of a mountain. The lift makes an angle of \(14.6^{\circ}\) with the horizontal. What is the change in the skier's gravitational potential energy?
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