A car whose weight is 20,0 N (mass is approximately 20 kg) is traveling at a constant speed of 25 m/s up an incline that makes an angle of 4 degrees with respect to the horizontal. What k-work does the gravitational interaction contribute to the car every second? Does this particular interaction transform energy to or from the cars kinetic energy? Explain
The answer is -340 J/s and from but I don’t understand how to get this. Any help would be appreciated
1 Answer
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First, resolve the velocity of the car in the vertical direction, v-vert = 25sin(4) = 1.7 m/s
[Since the maximum precision of the value of velocity given to us is 2 sig figs, (25), we can only quote v-vert to 2s.f as well. More on this below]
So now you know that the car moves upwards in a gravitational field by 1.7m every second. That means it must do work AGAINST gravity. See, if it were falling, then it’s speed would be increasing, so gravity would be doing POSITIVE work on it. But since this car is rising, gravity does NEGATIVE work on it.
So by the equation
Gravitational Potential Energy = mgh,
you can see that in one second, gravity does -mgh of work on the car every second. So the RATE of doing work is -mg(rate of change of h). But rate of change of h is 1.7 m/s which is the upward velocity. [the minus signs here are explained right at the bottom]
So the rate of doing work is -mg(v-vert).
Notice that “mg” is actually the weight of the car, 2N.
So rate of work done by gravity is -(2)(1.7) = -340 J/s or watts (W).
[If you use v-vert = 1.744m/s for instance, you get -34878 J/s which rounds not to 340 but 350
With that said, I was taught to preserve at least 2 more sig figs than required in the final answer when working forward from a partial answer, as long as you don’t have to ADD two partials together somewhere along the way (we’re multiplying here). That’s usually how it’s done. You only round at the final answer. But somehow that doesn’t fit 340 J/s here. Is 340 an official answer?]
Why is there a minus sign in -mgh and -mg(v-vert)?
Because if v-vert, which is upwards, is positive as we have taken it to be, then g has to be negative because the acceleration due to gravity is downwards and so opposite to v-vert.