At serve, a tennis player aims to hit the ball horizontally. W...

Question

At serve, a tennis player aims to hit the ball horizontally. What minimum speed is required for the ball to clear the 0.90-m-high net about 15.0 m from the server if the ball is 'launched' from a height of 2.30 m? Where will the ball land if it just clears the net (and will it be 'good' in the sense that it lands within 7.0 m of the net)? How long will it be in the air? See Fig. 3–50.

Tennis player serving a ball horizontally from 2.30 m height, aiming to clear a 0.90 m net, with distances marked.

Accepted Answer

Welcome back. Everyone in this problem. An engineer is testing a new lightweight drone by launching it horizontally from a platform to see how far it can travel without propulsion. There's a fence 5.5 m tall located 30 m from the launch platform. The drone is launched from a height of 8.4 m above the ground. What is the minimum horizontal speed required for the drone to clear the fence? Will it land safely in the designated area that starts 10 m beyond the fence and extends for another 20 m. Determine the total time. The drone will be in the air from the moment of launch to when it lands. We have a diagram representing the scenario with our drone or fence and our landing area. And for our answer choices, A says the minimum speed is 25 m per second that it lands before the designated area. And the total time in the air is 0.77 seconds. B says its minimum speed is 29.4 m per second. It lands safely in the designated area and the total time in the air is 1.5 seconds. C says the minimum speed is 39 m per second. It lands safely in the designated area and its total time in the air is 1.31 seconds. And the D says its minimum speed is 45 m per second. It overshoots the designated area and the total time in the air is two seconds. Now, let's make sense of the information we have here. So let's start with the first thing we need, we need to find a minimum horizontal speed required for the drone to clear the fence. Now, if we look at our diagram, our platform is 8.4 m or fence is 5.5 m high. So the distance that the drone will need to clear is going to be the difference between 8.4 and 5.5 m. So let's call that distance. Why? Now if we want to figure out what that distance is? Ok. So the distance to clear the fence, then we can find it by subtracting 5.5 m from 8.4 m. And when we do that, we get our distance y to be equal to 2.9 m. Ok? So that's the vertical distance that we need to clear. Now, if we're trying to find a minimum horizontal speed, let's ask ourselves, what do we know about speed and distance? Well, recall. Ok. And let's name number this as part one of the question, recall that speed equals distance. Divided by time. So in this case, the horizontal speed, the horizontal velocity is going to be equal to the horizontal distance, which we can let's just call that X divided by the time it takes. Now, the thing is we know that horizontal distance, we know that horizontal distance is 30 m, but we don't know what time it will take. So how can we figure out the time? Well, let's think about it. Is there any other way we can find the time? If we notice here, we know that we want to clear a vertical distance of 2.9 m. And the same time it takes in the vertical plane to clear that distance is the same time it would take in the horizontal plane. So we should be able to find the time based on the information or, or looking at it based on the information we have about the vertical distance. Now, let's think vertically. Do we know anything that can help us? Well, if we make a note of the variables, we want, we want the time vertically, we know the acceleration is going to be the acceleration due to gravity. So we know that's going to be G and we take it to be 9.8 m per second squared. OK. We know the distance why that we need to clear 2.9 m. OK. And we know that our drone is moving from rest so its initial velocity U is going to be equal to 0 m per second. So do we know any formula that relates to time acceleration due to gravity and distance and initial speed? Well, we do, we know that we have the kinematic equation that says why our distance equals the initial velocity? In this case, the initial velocity in the Y direction. Uh Let me say, uy, so that's our initial velocity multiplied by the time plus half a acceleration due to gravity. So in this case, GT squared, if we do some transposition here, that means Y is going to, this is going to be zero because zero multiplied by the time equals zero. So Y equals half GT squared, which means then that T squared equals two Y divided by G. And if we square root both sides, OK, that means T is going to be equal to the square root, the square root of two Y divided by G. So to find the time we can substitute our distance Y and the acceleration due to gravity because no, that means that the time is going to be equal to the square root of two multiplied by or vertical distance 2.9 m all divided by the acceleration due to gravity which is 9.8 m per second squared. Now, we can put this expression into our calculator. OK? And when we do that, we should get the time to be equal to 0.77 seconds. Now what does that mean? Well, no, we have the time it takes, I remember we said the time it takes to clear the vertical distance is the same as the horizontal distance. So we can use it in our formula to figure out our minimum horizontal velocity. Because now, OK, there are four, that means that the minimum horizontal velocity VX is going to be equal to our distance, 30 m divided by our time, 0.77 seconds. And again, when we put that into our calculator, we should get our minimum horizontal velocity to be equal to 38.96 m per second, which if we round it to the nearest second, nearest meter per second, that would be 39 m per second. So that's the minimum horizontal speed needed for our join to clear the fence. Next, let's try to figure out the landing distance, where is this thing going to land? So we have our minimum horizontal velocity and if we go back to our diagram, no, we need to find out where it's going to land assuming that it continues at that minimum horizontal speed or velocity. Ok. Now here notice that the fence is 30 m away and the landing area is 10 m away from the fence. So for our drone to go in the landing area, ok. Let me put that here in blue. So for our drone to go in the landing area, that means it's going to have to land between or it's going to, when it touches the ground, it would have to cover a horizon, a horizontal distance between 40 m. OK? And 60 m. So that's essentially what we're saying. So we want to figure out if the distance that it would travel would be between 40 60 m. Now, here's the thing about that again, let's ask ourselves, what do we know, what do we know about distance and speed? Because remember we already have the speed. So let's put that as part two of our question. So what do we know? But again, to know we're solving for our distance, the sea, if it's between 40 60 m and just like in our problem before we know that our distance is going to be equal to our speed multiplied by our time. But here's the thing, here's the thing, we don't know what time it will take for our drone to get there. But again, just like we had said before, we can figure out the time it would take using the same equation for vertical motion. The difference is no that our vertical distance that it's going to cover. Instead of being here, the difference between 8.4 and 5.5 is gonna be 8.4 m because it's going from the platform to the landing area which is on the ground. So now taking why driving for an egg? Swear so or rather yeah, selling for tea, solving for A T where our vertical distance is 8.4 m. OK? Then no, if we use our same formula that we had before the time T is going to be equal to the square root of two multiplied by 8.4 m. OK. Divided by the acceleration due to gravity 9.8 m per second squared. And again, this is the time it takes to get to the landing area. No, if we put this expression into our calculator, ok, then no, we should get the time to be 1.309 seconds. And for a time of 1.309 seconds, then the distance is going to travel is going to be equal to or horizontal speed. 38.96 multiplied by sorry, 38.96 m per second, multiplied by our time, which is 1.30 nine seconds. And when we multiply those, we get X to be equal to 50.99 9 m or approximately 51 m. So what does that mean where 51 is between 4060? So since X is greater than 40 m, but less than 60 m, right, then it lands safely. Ok. So it does land safely in the designated area, designated area. So now we know if it lands in the area, ok. We know that its minimum horizontal velocity is 39 m per second. The last thing we need to figure out is a total time in the air. Well, guess what? We don't need to do anything else because remember we needed that total time. It was in the air to find where it lands and that total time is 1.309 seconds. So, or horizontal speed is 39 m per second. Total time in the air is 1.309 or approximately 1.31 seconds. Ok. And it lands safely in the designated area. C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Projectile Motion

Kinematic Equations

Energy Conservation

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