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Problem 78 Medium Difficulty

A 292-kg motorcycle is accelerating up along a ramp that is inclined $30.0^{\circ}$ above the horizontal. The propulsion force pushing the motorcycle up the ramp is $3150 \mathrm{N},$ and air resistance produces a force of 250 $\mathrm{N}$ that opposes the motion. Find the magnitude of the motorcycle's acceleration.

Answer

5.03 $\mathrm{m} / \mathrm{s}^{2}$

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Video Transcript

to solve this question, we can apply Newton's second law in the direction off the movement. So the movement happens in this direction. Let me say that everything that is pointing there, it's positive, and everything that is pointing to the opposite side is negative as a consequence. Then applying Newton's second law in this direction, let me call it the X direction we get the following than that force is equal to the mass off the motorcycle times Its acceleration then in that force has three components. He's one which is a positive component 3150 minus these one, which is a negative component because it points the negative X direction. And then we have apart off the weight force because the weight force could be the composer in two components. These one and these one let me cold. These the acts component and is the y component. We care about the X component for now. So the X component acting there in the opposite direction after amusement. So it is negative. So miners. But will you x. This is because of the mass off the motorcycle times its acceleration. Then how can it? Coakley to the X component off. The weight force knows the following these angle here is started agrees. Right now, let me extend the weight force to form another triangle. These one, then this is a 90 degree angle before this must be a 60 degree angle. Now note that the Y component off the weight force makes 90 degrees. If the surface, then if they make 90 degrees on this 60 degrees, these must be 30 degrees. Then this angle is 30 day grace, too. Now, to calculate what is the X component, we can use the sign off 30 degrees because the sign off 30 degrees in that context is given by the opposite side off the triangle. I'm talking about this triangle. Sure, the opposite side is the X component off the weight force divided by the high party news, which is the magnitude of the weight force. Then the X component off the weight force is It goes to the magnitude of the weight force times this sign off 30 degrees, which is 1/2 So W X is he coast to know where you divide it. By true, the weight forces given by the mass off the motorcycle times acceleration of gravity near the surface off the earth. Then we have these equations. Also remember that G is approximately 9.8 meters per second squared, then going back to our equation. We have 3150 minus 250 miners. 292 times 9.8. Divided lecture is ecos to 292 times acceleration. Then we can solve this equation for the acceleration as the acceleration is the cost to this mind is this Miners D's times in 9.8, divided by True is 4.9 and these world thing is divided by 292. These results in an acceleration of approximately five 0.3 meters per second squared.