**Problem Statement:**

A boy throws a ball from the origin of coordinates (see figure). The ball has an initial velocity **v _{0}** (m/s) whose direction is α = 60

^{0}. The boy wants the ball to fall into the back of a truck (whose length is L = 15 m) moving at a constant velocity

**v**= 20

_{C}**i**(m/s). The initial distance between the truck and the boy is d

_{0}= 10 m.

Calculate:

- The minimum value of v
_{0}for the ball to fall into the back of the truck. - The maximum value of v
_{0}for the ball to fall into the back of the truck.

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**Solution:**

First of all we are going to write the position vector of the ball as a function of time. In order to do that, we integrate the acceleration vector to obtain the velocity and then integrate the velocity vector to obtain **r**. The acceleration of the ball is due to gravity:

The initial velocity vector of the ball in terms of its components is:

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

We obtain the velocity vector as a function of time integrating the acceleration:

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

Substituting the initial velocity vector and grouping:

Now we integrate the velocity to obtain the position vector:

The initial position vector r_{0} is null because the ball starts moving from the origin of coordinates.

Next we are going to write the two components of the position vector:

The truck is moving at a constant speed, therefore the x coordinate of its position as a function of time is given by:

You can see that we have used **the same system of coordinates** to write the position vector of both the ball and the truck.

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For the ball to fall into the back of the truck, the x-component of the position vectors of both must be equal at the same time:

The time in equation (4) can be calculated from equation (2). When the ball falls into the back of the truck, the y-component of its position vector is zero. Therefore:

And substituting the time in equation (5) into equation (4):

Substituting the givens of the problem statement into this equation and operating you will obtain a second degree polynomial. Its positive root will be v_{0} (you have to take the positive root because we are calculating the magnitude of a vector, which is always positive).

Finally v_{0} is:

This is the minimal value of v_{0} for the ball to fall into the back of the truck (at the rear end).

v_{0} will have its maximum value when the ball hits the front end of the back of the truck. In this case:

An using the same procedure as before:

Do not forget to include the units in the results.

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