The function describing the position (height) of the Aries as a function of time is given by

`Y(t) =A+B*t +C*t^2+D*t^3+E*t^4` (1)

The speed of a material point that moves according to a law `Y=Y(t)` is given by the first derivative:

`V(t) = (dY(t))/dt =B+ 2C*t +3D*t^2+4E*t^3` (2)

Also the acceleration of a material point that moves according to a law `Y =Y(t)` is given by the second derivative of the space with respect to time (or equivalent by the first derivative of the speed `V=V(t)` with respect to time):

`A(t) =(dV(t))/(dt) =(d^2Y(t))/(dt) =2C +6D*t +12E*t^2` (3)

a) The vertical acceleration of Aries as it lands is given by the combined forces of the moon gravitational attraction (which pulls the module downwards) and of the thrusters of the module (which push the module upwards. Thus the overall acceleration is

`a = (G_("moon")-F)/M =g_("moon") -F_("thrusters")/M`

As can be seen from the expression (3) above the acceleration is not constant over time. It is a second order function of time.

With the data in text one has

`A(t) =2*2.284 -6*0.2039*t +12*0.006419*t^2 =4.568 -1.2234*t +0.077028*t^2`

The graph of the acceleration is below. In absolute value the acceleration is positive (hence directed upwards) in the first 6 seconds of descend, then it is negative (directed downwards) from the 6th second to the 10th second when the vehicle lands.

b)

The vertical component of velocity in the first frame can be found by taking `t=0 s` is the expression (2) above.

`V(0) =B =-16.81 m/s`

c)

Since the mass of the Aries module is not given, one can not determine the magnitude of the gravitational force on it. However, the direction of the gravitational force is downwards toward the Moon.

d)

The free body diagram at different heights on Aries descend is in the figure below.

e)

The magnitude and direction of the total force on the module follows the same pattern as the acceleration does. Initially until second 6, it is positive upwards, then from second 6 to second 10 it is negative (directed downwards).

f)

Since the mass of the Aries module is not given one can not determine the magnitude of the thrust on it. However the direction of it upwards.

e)

Well, the technical adviser did a fairly good job. This is because the acceleration (and thus the overall force) on Aries is not constant but varies over time. Initially when the descend speed is high, the net force is pushing the module upwards, finally, when the speed of descend is small enough, a final pull downwards makes the Aries landing.

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