Position Velocity Acceleration vectors - Parabolic motion and free fall

Problem Statement:

A monkey drops a coconut from the top of a building of height y0 = 300 m without initial speed a day without wind. A hunter located at a horizontal distance x0 from the coconut wants to hit it. For the sake of simplicity we will assume that the bullet is initially at ground level. The initial velocity of the bullet has a magnitude v0 = 350 m/s and its direction is α = 300.

Position Velocity Acceleration vectors - Parabolic motion and free fall

Calculate:

  1. The position, velocity and acceleration vectors of the bullet as a function of time.
  2. The position vector of the coconut as a function of time.
  3. The time at which the bullet hits the coconut.
  4. The horizontal distance x0 between the hunter and the coconut.

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Solution:

As there is no wind, the only acceleration that the bullet has when it falls is that of gravity:

To calculate the velocity vector of the bullet as a function of time, we will first calculate the Cartesian components of the initial velocity vector:

In the figure above it you can see that the relations between the module of v0, α and the Cartesian components of the vector are:

Ans substituting the givens:

To calculate the velocity vector we make use of the definition of the acceleration vector:

If you want to see in more detail how this integral is done, please check Problem 1.

Substituting the components of the initial velocity calculated previously and grouping:

Note that the x component of the velocity vector will remain constant throughout the motion. The only component that varies is the vertical, since the motion described by the bullet is parabolic.


Now we calculate the position vector using the definition of velocity:

The initial vector position of the bullet is null since initially it is at ground level (at the origin of coordinates). We can write separately each component of the position vector:


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We will use the same procedure to determine the position vector of the coconut. It has no initial speed, and therefore v0 = 0

The initial position vector of the coconut with respect to the chosen coordinate system is:

Its acceleration is also g.

The coconut velocity as a function of time is given by:

And its position vector:

Therefore, the y component of the position vector for the coconut is given by:


When the bullet hits the coconut the y component of both positions vector are equal. Therefore we equate equations (2) and (3):


Finalmente, para determinar la distancia x0 entre el cazador y el coco sustituimos este tiempo en la ecuación (1):

Finally, in order to determine x0, ie, the distance between the hunter an the coconut, we substitute this time into equation (1):

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

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