1. In our Frank/Mary coordinate systems. Frank throws a rock...

1. In our Frank/Mary coordinate systems. Frank throws a rock at t = 0 from a position ($x_0$, 0) with a velocity u and at an angle $\theta$ above the positive x-direction. Mary's coordinate system is moving to the right at a constant velocity "v" relative to Frank's and the origins of the two systems coincided at t = t' = 0. Given [$x_0$, $u_0$, $\theta$, v], Calculate: a. The position (x, y) and velocity ($u_x$, $u_y$) of the rock relative to Frank's coordinate system as a function of time. Apply the Galilean transformations to Frank's calculations to obtain: b. The position (x', y') and velocity ($u_x'$, $u_y'$) of the rock relative to Mary's coordinate system as a function of time.

Transcript

00:01 So, in this question, we are given with a person standing above a cliff.

00:07 We can say this is a cliff of height h.

00:12 So, cliff height is h.

00:15 And the person standing on it is throwing a stone or rock horizontally with a velocity p.

00:26 So, this is the velocity of the stone being projected.

00:31 So, we can generalize this.

00:34 This will come and fall on ground like this.

00:37 So, we can generalize the path as a half projectile.

00:41 We can call this situation as a half projectile.

00:45 Now for sign convention, we can take the positive x direction as positive and negative x direction as negative.

00:54 And the positive y direction is positive and negative y direction is negative.

01:02 So, for origin purpose, we can take origin as the person above the cliff.

01:08 So, this point is our origin, which we can say 0, 0 or we can take as our reference point where x equal to 0 and y as well as equal to 0.

01:24 So, we can see since now considering the sign for both the quantities given to us, we can see the velocities along positive x direction.

01:36 So v and it is horizontal velocity.

01:40 So, we can write it vx which will be plus because this is positive because in the positive x direction also the height above the height h, we can assign directions also we can write this as i cap.

01:59 And for height h since it is in the negative direction of y.

02:09 So, we can write it as minus h or we can simply write it as h minus j cap.

02:19 So, now finding the source from this situation, finding the time taken for it to reach to the ground.

02:32 So, we can take another condition here.

02:35 Let us first see the diagram.

02:39 There is no horizontal acceleration, but there is vertical acceleration, which is acceleration due to gravity.

02:47 This is inbuilt acceleration or this is by default we can take it.

02:52 So this sign is also negative.

02:56 So we can separate the quantities along both the axis.

03:03 Along x axis we can write our initial velocity which is ux we can write which is plus v and the horizontal distance it will cover let us take it as t and this will be plus v also.

03:21 Similarly, no horizontal acceleration is present.

03:25 So we can write it as zero.

03:28 And along y, along y axis also we can represent the quantities along y axis.

03:37 This will be uy which will be again zero because there is no vertical acceleration, vertical velocity, initial velocity and the height covered by the rock will be minus h and the acceleration due to gravity which will be the vertical acceleration which will be minus g.

04:04 So, now for calculating the time taken by the rock to reach the ground, we can write the equation like this.

04:12 We can see along y we can what we can write along y we can write y equal to initial velocity multiplied with time plus half of vertical acceleration multiplied with time square...

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