Question

Two objects, objects X and object Y are held
together by a string light and are released from rest
near a planets surface in the orientation that is
shown in the figure. Object X has a greater mass
than object Y. A graph of the acceleration as a
function of time for the systems center of mass is
shown for the 4s. The positive direction is.
considered to be upwards. How does the speed of
object X Vx Compare to that of the systems speed Vs
after the objects have fallen for 4s?

Answer 1

Newton's second law allows us to find the results for the value of the acceleration and the velocity of the bodies are:

• The acceleration is constant for all bodies

• The speed increases linearly for the bodies and is the same for all.

Newton's second law says that force is directly proportional to the product of mass and acceleration

        F = m a

Where F is force, m is mass and acceleration.

As the body is close to the planet the force is the universal gravitational force

         $F = -G \frac{Mm}{r^2}$

Where F is the force, G the universal gravitational constant, M the mass of  the planet and m the mass of another body and r the distance between them

Let's substitute

          $-G \frac{Mm}{r^2 } = m a$

          $a = - G M/r^2 = - g$

We can see that the acceleration of small bodies (x or y) does not depend on their mass, the negative sign indicates that the force is directed to the center of the planet.

In a acceleration graph versus the time is the acceleration in constant for the two bodies and for their center of mass.

If we make a graph of velocity versus  time, it is given by

         v = v₀ - g t

This graph is a line of constant positive slope; the velocity values ​​for the two bodies are equal since the acceleration of gravity does not depend on the mass of the body.

In the attachment we can see a scheme of the graph of acceleration and velocity versus time

In conclusion using Newton's second law we can find the results for the value of the acceleration and the velocity of the bodies are:

• Acceleration is the same for all bodies

• The speed increases linearly for the bodies and is the same for all.

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