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The question I'm trying to answer is this:

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This is the solution:

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I can understand the first part of the solution when the car is on a straight line, but I'm havung difficulty with understanding the second lart of the solution. More specifically, I can't understand why the maximum speed would occur when there is no acceleration because, by the definition of acceleration, if it was accelerating up the slope it would increase its velocity until it reaches a maximum.

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    $\begingroup$ The car keeps accelerating until it reaches maximum speed, so maximum speed is reached when there is no further acceleration. $\endgroup$ – hdhondt Aug 7 '16 at 23:33
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The sentence used in the solution is a bit strange for me; specifically I think that the maximum speed will be the one driven on the flat surface and the speed will actually decrease going up the hill.
Conceptually, what they are calculating is clear though: there is an additional work rate to be done against gravity, i.e., the power output of the car needs to not only do the work against resistance but also against the gravity field of the earth, thus the second term.

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On the straight section of the path, there is no acceleration (constant velocity). The forces acting on the car in the level (horizontal) direction are balanced. So the 600N force directed to the left is balanced by a 600N force directed to the right.

Once the car reaches the ramp, there is an additional force due to gravity (392N) down the ramp in addition to the 600N force directed down the ramp at the bottom. On the ramp, the forces are unbalanced, and the car is actually being decelerated (in the downward direction) along the ramp. But the power (rate at which work is being done) is the same it was was on the level. So the force in the upward direction along the ramp changes from 600N to 992N. When the downward forces are balanced by the upward forces, the acceleration on the ramp will be zero and the velocity will be at its maximum. 992N is the max force that can be generated by the constant power at that point. It can't go any faster from that point on while its on the ramp.

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