Position Velocity Acceleration vectors - Changing reference frame

Problem Statement:

A boy kicks a ball from the top of a Cliff of width x0 = 10 m and height y0 = 15 m with respect to the reference frame O (see figure). The initial velocity of the ball in terms of its magnitude and direction is v0 = 15 m/s, α = 300.

Position Velocity Acceleration vectors - Changing reference frame

 

  1. Calculate the position, velocity and acceleration vectors for the ball as a function of time with respect to the frame of reference O.
  2. Write the same vectors with respect to the reference frame whose origin is O’ (see figure).
  3. Calculate the time at which the ball reaches the ground for each frame of reference.

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Solution:

The aim of this problem is to learn to express the vectors that describe the motion of an object with respect to different reference systems.

With respecto to O, the initial position vector of the ball is:

And its initial velocity:

Substituting the givens:

If you want to see in more detail how these components are calculated check out Problem 6.

The acceleration vector is given by:

Next we are going to use the definition of acceleration to calculate the velocity vector of the ball as a function of time.

In Problem 1 you can see in more detail how this integral is solved.

We use the definition of the velocity vector to calculate the position vector:

And using the velocity vector calculated previously:

Now we substitute the initial position vector of the ball and group terms:

And for later use we write the components of the position vector:


With respect to O’ the initial position vector r0 is zero because both the origin of the frame of reference and the ball are initially at the same position.

The initial velocity vector of the ball is the same as with respect to O.

Following the same procedure as before we obtain the vectors v and r as a function of time referred to O’.

We can write the components of the last one for later use:


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We calculate the time of flight of the ball with respect to O equating to zero the y – coordinate of its position given by equation (2):

To calculate the flight time of the ball with respect to O’ we have tu use equation (4). But this time the y’ – coordinate will be:

Because with respect to O’ the ground is y0 meters away but along the negative direction of the y axis.

Using equation (4) and finding the roots of the polynomial:

As you can see, the time of flight is the same for both frames of reference.

Do not forget to include the units in the results.

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