(II) During a workout, football players ran up the stadium sta...

Question

(II) During a workout, football players ran up the stadium stairs in 75 s. The distance along the stairs is 83 m and they are inclined at a 33° angle. If a player has a mass of 88 kg, estimate his average power output on the way up. Ignore friction and air resistance.

Accepted Answer

Welcome back. Everyone in this problem. A bicyclist travels 72 m up along a road inclined at an angle of 45 degrees with a horizontal on her bicycle, given that the total mass of the bicyclist and her bicycle is 95 kg. And the time she takes to travel the distance is 65 seconds. Calculate the average power output of the bicyclist. Assume that the friction and ir resistance are negligible for our answer choices. A says it's 1.4 multiplied by 10 watts. B 7.3 multiplied by 10 squared watts, C one multiplied by 10 cubed watts and D 4.7 multiplied by 10 to the fourth watts. Now, first, let's try to understand what's really going on here. So we're talking about a bicyclist traveling up an incline. Ok. And so far, OK. We know that this road is the length of this incline is 72 m. OK. Represent that here. And we also know if we were to put our bicyclist here and give our bicyclist a mass M, OK? Then we also know that the weight of the bicyclist would be m multiplied by the acceleration due to gravity G and we know that the bicyclist is inclined at an angle of 45 degrees. Let's let the be equal to 45 degrees. So if we make a note of all the information we have, we know that she is inclined at a 45 degree angle that the length are of the incline. Ok. Let me just get this out here. The length are the distance traveled that is, is 72 m. We also know that the time taken t is 65 seconds. And we know that the mass of the cyclist is 95 kg. And from it, we're trying to figure out the average power output of the cyclist. Now let's ask ourselves, what do we know about power? Well, recall, recall that power is the rate of work done, which as a formula is defined as work divided by time. Ok. Now the thing is we know the time it takes for the cyclist to travel, but we don't know the work that was done. So do we know anything else about work? Well, we also know that work, the work done is equal to the mass of an object multiplied by the acceleration due to gravity multiplied by the height that it has traveled through, which importantly, is it vertical height? Ok. Now, due to gravity being a conservative force, the path taken doesn't matter. So the work done along the incline is the same as the work done to lift the object to its final height from its initial height against the gravity. So let's let this height here to the top of our incline, from the, from the bottom of our incline to the top be equal to edge. OK. Now, if we think about it, we don't have a value for but or in Clan forms a right angle or sorry or in Clan forms a right triangle where is the vertical height? And our, our line here at the bottom of our diagram is the horizontal. So they form a 90 degree. OK. And that means at that point, we can use what we know about right triangles to help us figure out the value of H notice that H is opposite to our anger theta. OK? And we already know what the value of R is the high pou. So in trigonometry, what do we know that relates the opposite side to the high pou? Well, we know that it's the sine ratio. So if I make a quick note here, we know that the sine of an angle equals the opposite side divided by the high party. So in our problem here, the sine of theta is going to be equal to the opposite side H divided by the Hyperten R if we multiply both sides by Rh equals R sine theta. So now we have a way to find our vertical height H OK, because it's R sine theta. So really then the work done is actually equal to MG multiplied by R sine theta. And if we were to substitute this formula for the work done into our formula for power, we'll put that in red here. Then that means the poor is going to be equal to the work mgr sine theta. OK. Div divided by the time it takes t we know all of these values. So now that we have them, we can go ahead and solve for the power. We know that the cyclist's mass is 95 kg. Let's state acceleration due to gravity to be 9.81 m per second squared. OK. Oops. And we know that the distance traveled R is 72 m multiplied by the sine of the angle formed between the incline and the horizontal, which is 45 degrees. And all of that will be divided by the time it takes for the cyclist to cover that distance, which is 65 seconds to figure out the power output, we can put this expression into our calculator. Now, when you do that, you should find that the power output is 729.9 oops 0.956 watts. Now, if we write that to two significant figures, writing it in standard form is going to be 7.3 multiplied by 10 squared watts. So in other words, the average power output of the bicyclist, assuming that friction and air resistance are negligible is 7.3 multiplied by 10 squared watts. If we look back at our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

(II) During a workout, football players ran up the stadium stairs in 75 s. The distance along the stairs is 83 m and they are inclined at a 33° angle. If a player has a mass of 88 kg, estimate his average power output on the way up. Ignore friction and air resistance.

Key Concepts

Work and Energy

Power

Inclined Plane

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