To find the power output of a machine, we use the definition of power, which is the rate at which work is done. It is calculated by dividing the total work done by the time taken to do that work.
Power (\(P\)) is defined as work done per unit time.
\[ P = \frac{W}{t} \]The unit Joule per second (\(\text{J/s}\)) is equivalent to Watt (\(\text{W}\)).
\[ P = 200 \text{ W} \]Thus, the power output of the machine is 200 W.
Correct Option: C) 200 W
To determine what happens to the kinetic energy when the speed of an object doubles, we need to recall the formula for kinetic energy and analyze its relationship with speed.
B) It quadruples. As derived, when the speed of an object doubles, its kinetic energy becomes four times its original value.
To calculate the average power output, we need to determine the work done by the car and the time taken. The work done can be found by calculating the change in kinetic energy, as the car accelerates from rest. Power is then the work done divided by the time taken.
We are not given the mass of the car. In such problems, it's often implied that we should consider a unit mass or that the question is flawed. However, if options are given in Watts, it implies a specific mass is either assumed or can be derived. Let's assume the question implicitly refers to a car of mass \(m = 1 \, \text{kg}\) for calculation purposes, or there might be an error in the question if a specific mass is not provided and the answer is a fixed numerical value. Given the options, let's re-evaluate. If the question intended to ask for power per unit mass, the unit would be W/kg. Since the options are in Watts, there must be a missing piece of information or an implicit assumption. Let's assume a common scenario where a mass is implied or the question is a conceptual one where the numerical value is expected to be derived from the given numbers in a specific way. However, without mass, we cannot calculate kinetic energy or power. This indicates a potential flaw in the question as presented, or a common competitive exam trick where a standard value is expected. Let's assume a mass of \(1 \, \text{kg}\) to proceed, as it's the only way to get a numerical answer in Watts.
The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy.
\[ W = \Delta KE = \frac{1}{2}mv^2 - \frac{1}{2}mu^2 \]\[ W = \frac{1}{2}m(10 \, \text{m/s})^2 - \frac{1}{2}m(0 \, \text{m/s})^2 \]\[ W = \frac{1}{2}m(100 \, \text{m}^2/\text{s}^2) \]\[ W = 50m \, \text{J} \]Average power is defined as the total work done divided by the total time taken.
\[ P_{\text{avg}} = \frac{W}{t} \]\[ P_{\text{avg}} = \frac{50m \, \text{J}}{5 \, \text{s}} \]\[ P_{\text{avg}} = 10m \, \text{W} \]Since the options are numerical values in Watts, and we have \(10m \, \text{W}\), this implies that \(m\) must be \(10 \, \text{kg}\) for option B to be \(100 \, \text{W}\), or \(m\) must be \(1 \, \text{kg}\) for option C to be \(10 \, \text{W}\). This is a common issue in competitive exams where mass is sometimes omitted, and one must infer it from the options or assume a standard value. If we assume the question expects a specific numerical answer from the given values, and option B (100 W) is the correct one, then \(10m = 100\), which means \(m = 10 \, \text{kg}\). Let's proceed with this assumption.
If we assume \(m = 10 \, \text{kg}\):
\[ P_{\text{avg}} = 10 \times 10 \, \text{W} = 100 \, \text{W} \]B) 100 W — By assuming a mass of 10 kg (which is necessary to arrive at one of the given numerical options), the average power output is calculated as 100 W. This is a common way such questions are framed in competitive exams when a critical piece of information (like mass) is missing, and one of the options is expected to be the answer.
The question asks to identify the energy source used in a hydroelectric power plant. We need to understand the basic principle of how hydroelectric power is generated.
C) Water flow is the correct answer because hydroelectric power plants harness the energy from the movement of water, usually from a dammed river or a reservoir, to generate electricity.
This problem involves the fundamental relationship between power, work, and time. Power is defined as the rate at which work is done or energy is transferred. We are given the power output of a car engine and the duration for which it operates, and we need to calculate the total work done.
Correct Option: A) 5000 J is the correct amount of work done by the engine in 10 seconds.
The question asks about the relationship between power and work done. To answer this, we need to recall the fundamental definition of power in physics.
B) Power is directly proportional to work done. This statement is directly derived from the definition of power \(P = W/t\). If time is constant, as is often implied when comparing work done, then power is directly proportional to work done.