A ball is thrown toward a cliff of height h with a speed of 30 m/s and an angle of 60° above horizontal. It lands on the edge of the cliff 4.0 s later. c. What is the ball's impact speed?

Video transcript

Hey, everyone in this problem, we have Stephen who lives in the second floor of an apartment building. And one day when he's leaving the house, he accidentally takes this pouch with the house keys in it. Ok? He realizes this when he reaches the parking lot in front of the building and he's gonna throw that pouch up to his son on the balcony. Yeah, he's gonna throw it with a speed of 25 m per second at an angle of degrees and his son is gonna catch it after three seconds and we are asked to calculate the impact speed of the pouch, ok? So that speed as the sun catches it. And we're told to consider that the balcony has a height of H. We're given four answer choices here. Option A 8.3 m per second. Option B 24 m per second, option C 5.3 m per second and option D 25 m per second. And we're gonna start by drawing a little diagram. So we have Stephen, ok? And he's got his keys and he is gonna throw them at 25 m per second at an angle of 75 degrees with the horizontal. And they're gonna go up to his son who was standing on the balcony and he is going to catch them. This balcony is located at a high itch and we wanna know this impact speed of those keys. Now let's write down the information we have and we need to consider both the X direction in the wide direction. So in the X direction, we have that V knot in the X direction. Well, it's gonna be 25 m per second, that initial speed. But then we have to account for the angle and we're gonna take up into the right to be positive. If we're looking at the X component or the horizontal component, this is gonna be related through cosine because it's the adjacent side. So we get 25 m per second multiplied by cosine of degrees. Now there's no horizontal acceleration A A X is equal to zero m per second squared. We're assuming we can ignore air resistance. And so what that tells us is that the final velocity in the X direction is also equal to cosine of 75 degrees multiplied by 25 m per second. OK. It's the exact same. And if we approximate that that is 6.4705 m per se, now we're looking for the impact speed. So we need the X component and the Y component and then we can calculate the speed. So we have the X component of that final speed and we are gonna come back to that in a minute. So we're gonna put a green box around that. So we don't lose track of it. We're gonna flip over to the Y direction. Now in the Y direction, the initial velocity of V not Y OK. This is gonna be 25 m per second multiplied by sign that 75 degrees and very similar to the X direction except in the wide direction we're talking about the opposite side. So we're using sign of the ankle. We wanna find that final why velocity? So that we can calculate the total impact speed. We know that there will be acceleration due to gravity acting here that's gonna act downwards. And so it's gonna be a negative acceleration negative 9. m per second squared. Delta Y is gonna be the height of that balcony. H OK. That's the distance that, that poach travels. And we're told that the sun catches the pouch after three seconds. And so the time that this takes is going to be equal to three seconds. If we look at our Y direction information, we have three known values V, not Y A Y and T. Hey, that's enough to solve for 1/4 variable. We wanna find V F Y. So let's solve for it. We're gonna choose a kinematic equation that has those four variables we're gonna ignore delta Y because we don't know that value of the height of the balcony. And that's not what we're interested in finding. So we get that V F Y is equal to V not Y plus A Y T. OK. So the final velocity is gonna be equal to the initial velocity m per second, multiplied by cosine whoops, not cosine sine of degrees plus the acceleration negative 9.8 m per second squared multiplied by the time it takes three seconds. OK? If we simplify everything on the right hand side here, we are gonna get a final velocity of negative 5. m per second for the vertical component. Now, that might look a little confusing because it's negative. But let me go back to our diagram and explain why that makes sense. OK? When Steven throws this up to his son, OK. That pouch is gonna go up just above the sun and then come back down as he catches it. OK? It's gonna go up and then he's gonna catch it on the way back down. And so on the way back down, that vertical velocity is gonna be pointing downwards. And so that's gonna be a negative value. OK? That's why we have that negative there. So we wanna calculate the impact speed. So we care about the magnitude of that total speed. We know that we have a positive horizontal component. So we have the horizontal component pointing to the right of 6.4705 m per second. We have a vertical component that's pointing down because it's negative. It has a magnitude of 5.25185 meters per second. And what we want to calculate is the hypotenuse that connects these two. OK. That is gonna be the final speed we have. So the final speed V F squared using Pythagorean theorem is gonna be equal to V FX squared plus B F Y squared. I've put the absolute values here just to indicate that we wanna take the magnitude of each of those values. In reality, we're squaring it. So whether you use positive or negative, it's gonna end up positive anyways. Um So it's OK if you don't include those there, OK? So V F squared is going to be equal to 6.4705 m per second squared plus 5.25185 m per second. All squared simplifying on the right hand side, we get V F squared is equal to 69.4493. We have meters squared per second squared. OK? Because we had meters per second and then we squared it and we're gonna take the square route and when we take the square route, we're gonna take just the positive route. OK? Or the magnitude because again, we're looking for speed, we don't care about the sign and we get that this is equal to 8. meters per second. And that is that impact speed of the couch or sorry, the pouch containing the keys when the father throws them up to the sun. All right, if we compare this to our answer choices, these are rounded to two significant digits. So we can see that the correct answer is option a 8. m per second. Thanks everyone for watching. I hope this video helped see you in the next one.