A ball is thrown vertically upward from the ground with a velocity of 30m/s (a) How long will it take to rise to its highest point? (b) How high does the ball rise? (c) How long after projection will the ball have a velocity of 10 m/s upward? d) Of 10m/s downward? (a) 3.06 s b) 45.9 m c) 2.04 s d) 4.08 s

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Hi there. So for this problem we are told that a ball is thrown vertically upward with an initial velocity from the ground of course, with initial velocity B initial one That is equal to 30 m/s. So the question for part A of this problem is how long will it take to rise to its highest point? No, we know that this bowl is going to produce something like this, he's going to get to the highest point and then it's going to fall again to the ground. Now in the highest point, its velocity, that means its final velocity is equal to zero momentarily. So what we can use is the equation from Kingdom Matters that it states that the final speed is equal to the initial speed minus the acceleration, which in this case is the acceleration due to gravity times the time. So, since the at that point, since the final speed is equal to zero, we just simply solve for the time. Then from this we just can pass um this term to their left side and then divide the initial velocity by the acceleration due to gravity. And now we substitute the numerical values, we know That the initial speed is 30 m/s and the acceleration due to gravity is 9.8 m/s square. So from this we obtain a value of 3.06 seconds. So that's the solution for part A of this problem. Now, for part B, we are told now, we are asked about how high does the ball rise and now again, what we need to obtain is the height at this point at its maximum height. So to do that, we use another equation from Kitimat. It's that is that the final speed square is equal to the initial speed square minus two times the acceleration. Again, the acceleration due to gravity times the um difference in the vertical position. Now, in this case against this we wanted when it is at its maximum height, then the final speech should be equal to zero. So if we sold for the maximum height we will obtain that. This is just simply the initial speed to the square divided by two times the acceleration due to gravity. And now we substitute those values 30 m per second to the square and it's divided by two times 9.8 m per second square. So using our calculator, we obtain a value of 45.9 meters. So that's a solution for party of this problem. Now, for parsing, we are asked about how long after Projection will the ball have a velocity of 10 m per second upward. So in this case we are given that the final speech should be equal to 10 m/s and we need to find the time at which this happens. So in this case we use the same equation as before that the final speed is equal to the initial speed minus two times um sorry minus the acceleration due to gravity times the time. And now we just need to simply solve for the Time again. So in this case we're going to have that this is the final speed minus initially speed divided by minus acceleration due to gravity, and this is going to be equal to the time. Now in this case we substitute those values, the final speed is 10 meters per second, the initial speed is 30 m per second and this divided by the acceleration due to gravity 9.8 m per second square and the minus in here. So from this we obtain a value of 2.04 seconds. So that's a solution for this problem.

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