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A car's engine must exert a force of $2,000 \mathrm{N}$ to maintain a speed of 30 $\mathrm{m} / \mathrm{s}$ up an incline. What is the power provided by the engine during this motion?(A) 20 $\mathrm{N}$(B) 40 $\mathrm{N}$(C) 80 $\mathrm{N}$(D) 160 $\mathrm{N}$

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Chapter 4

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in sector size. We have a car that is coasting in neutral, meaning that no power is being given by the engine to the car at a speed of 20 m per second and down a hill that has an inkling on inflammation off two degrees with the horizontal, the car has a mask off 1500 kg in question. A. We have to suppose that the car is now moving in a level road and we want to completely, What is the power that the engine must give to the car such that it travels at the same speed of 20 m per second? Well, first of all, uh, what we need to consider in this exercise is that when the car is closing down the road, the forces that act upon it are that traditional floors that causes an acceleration parallel to the inclined plane off MG science data and and we have also have and their resistance force, that I'm gonna call f air the points of works. Now if the car is moving with a constant speed, that is, the car reached a terminal speed, then the some of the forces must be zero. So you know that f air, the force, The force of the answers over the car minus mg sign data is equal to zero. So from here we can obtain that the force of the air exterior toe over the car is mg time Science data and theatre here is just to okay Now when the car is moving at the left in level road, we also have the force off air the air resistance we do not have the The Tarallo judicial component of the gravitational force against force is just pointing downwards and not causing an ex exploration on the car. This means that the force that is making the car go forward is the force that the the the engine of the car is providing it. And if the car is moving with a constant speed of 20 m per second, then the force that the car the engine of the car exerts over the car is equal to it was our mind is a force of the force of resistance of air is equal to zero. This means that the forced the engine exerts is equal to the force that the characters and as we saw before this is equal to mg. Time to sign of to no. Now if we want to calculate what is the average power that's given by the the engine? And we know that the power is equal to Theo. Energy that's given divided by the time of the energy is just the work that the the engine does. And this is equal to the force of the engine times the distance, the divided by t. Now I noticed that the cars movie under at a constant speed, so D divided by t is just the speed. So we have the force of the engine times this Pete, this is N g sign, afraid of times the speed V so I can calculated explicitly. M is 48 kg g is 9.8 m per second square. Then we have multiplied by sine of two degrees men by 20 m percent. So the power is equal to 10 to the fourth lots or 10 kilowatts. This is the power that must be provided by the engine in question be We have to suppose that the power that the engine gives to the car it's even which of 40 kilowatts and with this information. We want to know what is the steepest incline. So this means that we want to calculate the angle data that the car can climb up at a constant speed of 20 meters per second. For this case you have again here inclined Twain. We want to complete this data here we have the car that subject to the following forces. Actually, the cars moving imports. I'm going okay with us. Um car subject to the falling forces. There is the gravitational force Tangential component MG science. Later, there is the force of the engine experience over the car moving pointing upwards. So this is the force of the engine and we also have the the force of the of the air resistance. So that's pointing upwards because of the car is moving. Actually the force of the air resistance pointing downwards because the car is moving upwards and for the car to move other at a constant speed of 20 m per second. Then all the forces must sum to zero. So have that the force of the engine most the force of the air minus MGI Sign of fada is equal to zero Now. What is the force of the engine gives to the car. As we saw before the power, he is equal to the force of the engine times V. Now the power is 10 is for 40 times into the third. This is equal to a of the force of the engine times 20 everything years in the international system of units. So he gets that the force of the engine is 2000 uniqueness. And we know how much. Uh, but we know what is the force of the air? Ah, for the for 20 m per second. So it can substitute everything here. So you have the force of the engine. Mine is the force of the air which, as we saw a Zen G times the south to then miners mg times a sign of data. It is what we want to find out now. F off the engine is 2000 unions. They have to subtract 15,015 1500 times of course times 9.8 m per second squared times sign of to and this is equal to I m g 10 sign of thieving. So this is 2000 new homes minus five 100 Newtons divided by M G, which is 1500 kilograms times 9.8 m for seven square Mrs Sign of Fada. So Fada is the arc sine of 1500. Newton's divided by 1500 kg. Time is 9.8 m per second square is the ark sign of 1/9 0.8, which is equal to 5.8 the police and this concludes there exercise.

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