The question in Spanish reads:

Problem 14: "Dos móviles parten simultáneamente del origen de coordenadas, ambos con MRUA y en la misma dirección. A los 5 segundos de la partida, la distancia entre ambos es de 50 m. Calcular la aceleración del segundo móvil, sabiendo que la del primero es 3 m/s². (Rta: 7)."

Translation and Problem Analysis:

Two objects start moving simultaneously from the origin with uniform acceleration in the same direction. After 5 seconds, the distance between them is 50 meters. We need to calculate the acceleration of the second object, given that the acceleration of the first object is 3 m/s². The answer is expected to be 7 m/s².

Solution:

1. Let:

   • $a_1 = 3 \, \text{m/s}^2$ (acceleration of the first object),
   • $a_2$ (acceleration of the second object, which we need to find),
   • $t = 5 \, \text{s}$.

2. Since both objects start from the origin (initial velocity and position are zero), we can use the equation for the position under constant acceleration: $x = \frac{1}{2} a t^2$

3. The position of the first object after 5 seconds is: $x_1 = \frac{1}{2} a_1 t^2 = \frac{1}{2} \times 3 \times (5)^2 = \frac{1}{2} \times 3 \times 25 = 37.5 \, \text{m}$

4. The position of the second object after 5 seconds is: $x_2 = \frac{1}{2} a_2 t^2 = \frac{1}{2} \times a_2 \times 25 = 12.5 a_2$

5. The distance between the two objects after 5 seconds is given to be 50 meters, so: $|x_2 - x_1| = 50$

6. Substituting the values: $|12.5 a_2 - 37.5| = 50$

7. Solving for $a_2$: $12.5 a_2 - 37.5 = 50$ $12.5 a_2 = 87.5$ $a_2 = \frac{87.5}{12.5} = 7 \, \text{m/s}^2$

Answer:

The acceleration of the second object is $a_2 = 7 \, \text{m/s}^2$.

Would you like further explanation on any part of the solution?

Additional Questions:

1. How would the solution change if the initial velocities were not zero?
2. What if the two objects started at different positions instead of the origin?
3. How does relative acceleration affect the distance between two objects moving in the same direction?
4. Can this problem be solved graphically? If so, how?
5. What would happen if the accelerations of both objects were equal?

Tip: When solving problems with uniform acceleration, remember that distance covered depends on both the time squared and the acceleration magnitude.