The 2-Mg car increases its speed uniformly from rest to 25 m / s in 30 s up the inclined road. D

The 2-Mg car increases its speed uniformly from rest to 25 m...

Key Concepts

Kinetic Energy

Kinetic energy is the energy that a body possesses due to its motion and is given by the formula (1/2)mv². In problems involving acceleration from rest, calculating the increase in kinetic energy helps determine the work done on an object, which is directly related to the power required for that change in energy.

Uniform Acceleration

Uniform acceleration refers to the constant rate of change of velocity over time. This concept allows one to use kinematic equations to determine both the final velocity reached and the displacement, which in turn facilitates calculations of kinetic energy changes and power requirements over the given time interval.

Power

Power is defined as the rate at which work is performed or energy is transferred over time. It can be considered both as an average value over a time interval and as an instantaneous value, often referred to as maximum power. When analyzing mechanical systems, distinguishing between average power and maximum power is crucial for understanding performance demands.

Engine Efficiency

Engine efficiency is a measure of how effectively the engine converts the input energy into useful mechanical work. An efficiency less than 100% means that some energy is lost, typically as heat or friction, so the actual energy input must be higher than the mechanical work output. This parameter is key when calculating the true power required from the engine to meet performance goals.

Work-Energy Principle

The work-energy principle states that the net work done on an object is equal to its change in kinetic energy. This principle is fundamental in mechanics as it provides a direct link between the forces acting on an object (and the work they do) and the resulting motion, thereby allowing one to determine energy requirements and power outputs needed for acceleration.

Transcript

00:01 In this problem, we have a car that's moving up in an incline with a constant speed 25 meters per second, and it achieves this speed in 30 seconds.

00:15 We want to find the power that this engine supplies during this time frame.

00:21 We're also given the efficiency of this engine, which is 0 .8, and the mass of the bar, which is 2 times 10 to the 6 grams.

00:32 Or divide by a thousand to get 2 ,000 kilograms.

00:40 When we're dealing with engines and power, it's important to know that the efficiency of that engine is equal to p -out divided by p -n.

00:52 P -out is the power that the engine exerts.

00:56 P -n is the power that the engine supplies, and this is what we're trying to solve.

01:02 So solving for pn, we get pn is equal to p out divided by epsilon the efficiency of the engine.

01:16 So we know that p out is equal to the force that the engine supplies times the velocity of the car.

01:28 We're given the velocity, but we don't know the force that the engine supplies.

01:33 So to do that, we draw a free body diagram of the car.

01:38 The car is at an incline so that means the weight of the car is straight down and we know that it has components so you have w y in the y direction and opposite that is a normal force fn and we also have w x and also have to the right the force that the engine exerts so the sum of the forces in the x direction we get the force of the engine minus the weight of the car, which is equal to mass and acceleration.

02:30 So we solve for, we get ma plus wx.

02:41 So in this equation, we don't know the acceleration of the car, and we need to find a value for the weight in the x direction.

02:50 Well, for acceleration, we know that that's just velocity divided by time.

02:58 And we're given both pieces of information.

03:01 Velocity is 25, and the time is 30 seconds.

03:06 So the acceleration of the car is 0 .83 meters per second squared.

03:16 And if you go back to the diagram, you can say sine theta, so this angle right here, is equal to wx divided by w.

03:33 So then for wx, we get the weight of the car times sine theta.

03:44 So if we put both in, we get f is equal to w sine theta plus m.

03:59 But we don't know the angle of this incline, but we're given a nice little triangle to find that angle.

04:09 So this is a triangle from the diagram, so 10 at the bottom one on the side...

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