Relative Motion - Rain and car problem

Problem Statement:

A car moves along a road with constant velocity vC measured with respect to an inertial frame of reference O at rest (see figure). There is no wind. The driver O’ observes that a rain drop falls with an initial velocity: v’0 = – 30 i – 25 j (m/s). With respect to O the initial velocity of the rain drop is given by: v0 = – 25 j (m/s). The rain drop is initially at a height h = 20 m from the ground.

Calcular:

  1. The velocity v of the car with respect to O.
  2. The acceleration of the drop with respect to both frames of reference.
  3. The time it takes for the drop to hit the ground with respect to both observers.
  4. What trajectory does the drop describe with respect to O? And with respect to O’?
  5. The velocity of the drop when it hits the ground.

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Solution:

In order to find the velocity of the car we are going to apply the Galilean transformations for the velocity to the initial velocity of the rain drop.

In this problem v and v’ are known and we must calculate V, that is, the velocity of O’ with respect to O.


Since both frames of reference are inertial, the rain has the same acceleration (gravity) for both of them:


With respect to O, the rain drop has a free fall motion with an initial velocity v0 = – 25 j from a height h = 20 m from the ground.

The equation for the position in a free fall motion is given by:

Notice that the initial velocity of the drop has a minus sign because it moves in the negative direction of the y axis.

Substituting the givens:

When the drop hits the ground the y of its position is zero. Substituting into the corresponding equation and taking the positive root of the polynomial:

The time it takes for the rain drop to hit the ground is the same for both frames of reference because in Classical Mechanics time is the same for all reference frames.

With respect to O’ the rain drop follows a parabolic trajectory because its initial velocity with respect to it has both horizontal and vertical components, and its only acceleration is that of gravity.


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With respect to O the final velocity of the rain drop is given by:

And substituting the flight time:

Where the minus sign means that this velocity is on the negative direction of the vertical axis.

We can use the Galilean transformation to calculate the velocity of the rain drop with respect to O’:

In this problem we have used g = 10 m/s2. Do not forget to include the units in the results of the problems.

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