Example 11 (Trajectory) The gravitational acceleration on a...

Example 11 (Trajectory) The gravitational acceleration on a planet has a magnitude gm/s². A bullet is fired from a point h?m above the ground with an initial velocity v??m/s, where (a) g = 2, h? = 12, v?? = ?10, 10, 1? (b) g = 6, h? = 10, v?? = ?8, 6, 1? Find the quantities below. 1. The time of flight T. 2. The range R. 3. The maximum height H. Solution: Set the origin on the ground directly beneath the

Transcript

00:01 All right, hello, in this question, we're told that we have a bullet that has fired some speed, v0, and it's fired from some distance, h, above the ground.

00:07 And it's going to experience acceleration due to gravity.

00:10 So it's going to have a parabolic trajectory before it hits the ground.

00:13 We're asked to find the range, r.

00:16 So that's the distance that it actually travels from its point of origin.

00:20 The maximum height, h.

00:22 So at the top of its trajectory, we're going to have some height, h.

00:26 And then t is also something we want to know.

00:29 And that's the travel time total.

00:31 Now, in our given a and b values, which we'll plug in at the end, we're given that there's a z component to the velocity.

00:38 And so right as i've drawn it here, this is two dimensional.

00:41 We have positive x, positive y.

00:43 Z is going to be coming into and out of the page.

00:46 And the fact that exists does not matter, because it doesn't change how long the flight time is going to be.

00:54 The only thing that dictates how long this is going to be is how high it goes.

00:57 So what its initial speed in the y direction is.

01:00 And its range is going to be dictated by just the x component velocity.

01:05 You can think of it, if we look down from above, this is our bullet.

01:08 Regardless of what direction in z we fire, it's going to travel the same distance, r.

01:15 And z is independent of the height dimension as well.

01:17 So we just don't have to worry about the z dimension here.

01:21 That's note, because we're going to use kinematics to do this, and we don't want to have three dimensions for kinematics.

01:28 We'll just use two, and that will be totally fine.

01:30 So in order to figure out all these values, if we're given the initial speed, we're given the initial height, and we're given the acceleration due to gravity, then we can use some kinematics equations.

01:41 We know that our speed in the x direction is going to be our distance we travel in the x direction divided by our time t, which in this case is going to be big t.

01:50 And i know what my speed in the x direction is.

01:52 It's just the x component of my speed.

01:54 And i know that that's a constant.

01:55 Ok, well, i have that.

01:57 How am i going to find what t is? well, i know that my position, my sum final height is equal to my initial height plus my initial speed in the y direction times time plus 1 half times my acceleration times time squared.

02:13 Now, i know that the total time t is going to correspond when i start here and end here.

02:18 And so i'm going to have my final height.

02:20 That's going to be 0...

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