Raindrops make an angle theta with the vertical when viewed through a moving train window. If the speed of the train is v (subscript t), what is the speed of the raindrops in the reference frame of the Earth in which they are assumed to fall vertically?

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Bentley Leach · Accepted Answer

Here Vrg = velocity of rain with respect to ground Vtg = velocity of train with respect to ground = Vt Vrt = velocity of rain with rspect to train Choosing the X direction as the direction of thetrain we have Vrg=Vrt+Vtg where the velocities are vectors, from the figure: tan⁡θ=VtgVrg Therefore, Vrg=Vttan⁡θ

Jeffrey Jordon · Accepted Answer

If θ is the angle the rain makes with the vertical (from the train&#039;s reference frame),   tan⁡θ=(velocity of train)(velocity of raindrops in Earth frame)

star233 · Accepted Answer

To solve this problem, let&#039;s consider the motion of the raindrops from two different frames of reference: the moving train frame and the stationary Earth frame.In the moving train frame, the raindrops appear to be falling at an angle theta with respect to the vertical. Let&#039;s denote the velocity of the raindrops in this frame as vrain (subscript t) and the velocity of the train as v (subscript t).Now, let&#039;s determine the relationship between the velocity of the raindrops in the train frame and the velocity of the raindrops in the Earth frame.In the train frame, the vertical component of the raindrop&#039;s velocity is given by vrain(subscript&amp;nbsp;t)*sin(theta), since it is the component of the velocity in the direction perpendicular to the train&#039;s motion. The horizontal component of the raindrop&#039;s velocity in the train frame is v (subscript t), since it is the component of the velocity in the direction parallel to the train&#039;s motion.In the Earth frame, the raindrops are assumed to fall vertically. Therefore, the vertical component of the raindrop&#039;s velocity in the Earth frame is simply the speed at which the raindrops are falling, which we&#039;ll denote as vrain (subscript e). The horizontal component of the raindrop&#039;s velocity in the Earth frame is zero, since there is no horizontal motion in this frame.Equating the vertical components of the raindrop&#039;s velocity in the train and Earth frames, we have:vraint·sin(θ)=vraineSince the horizontal component of the raindrop&#039;s velocity in the Earth frame is zero, we have:vraint=0Simplifying the equation, we find:vraine=vraint·sin(θ)Substituting vraint=0, we get:vraine=0Therefore, the speed of the raindrops in the reference frame of the Earth, in which they are assumed to fall vertically, is zero.

xleb123 · Accepted Answer

Let vt be the speed of the train and θ be the angle the raindrops make with the vertical when viewed through a moving train window.In the reference frame of the train, the raindrops appear to fall at an angle θ with respect to the vertical. However, in the reference frame of the Earth, the raindrops are falling vertically. We can use this information to find the speed of the raindrops in the reference frame of the Earth.Let vr be the speed of the raindrops in the reference frame of the Earth.The velocity of the raindrops in the reference frame of the train can be broken down into two components: one along the vertical direction and the other perpendicular to it. The vertical component of the velocity remains the same in both frames of reference, as the raindrops are falling vertically. However, the horizontal component of the velocity changes due to the motion of the train.The horizontal component of the velocity of the raindrops in the reference frame of the train is given by vt, as the train is moving horizontally.Since the angle θ is formed between the direction of motion of the raindrops (vertical) and the line of sight (horizontal), we can use trigonometry to relate the velocities in the two frames of reference:tan(θ)=vtvrSolving for vr, we have:vr=vttan(θ)Therefore, the speed of the raindrops in the reference frame of the Earth is given by vttan(θ).

Raindrops make an angle theta with the vertical when viewed through a moving train window. If the speed of the train is v (subscript t), what is the speed of the raindrops in the reference frame of the Earth in which they are assumed to fall vertically?

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