Answer:
Equation of motion
x(t) = -9.81*t + 3000
The object will strike the ground in 305.81 sec
Step-by-step explanation:
a) Determine the equation of motion of the object
If x(t) repsent the distance the object has fallen in t seconds, then we have x'(t) and x''(t) represent the speed and acceleration respectively.
Since the force due to air resistance is proportional to the velocity of the object with proportionality constant b = 50 N-sec/m, we have
<em>F(t) = 50x'(t) </em>(1)
Due to Newton's 2nd law of motion (assuming the mass remains constant
)
<em>F(t) = mx''(t) = 50x''(t)
</em>
As the mass was released from rest, <em>the acceleration is the gravity</em>, and
<em>F(t) = 50*(-9.81) = -490.5 N/m
</em>
The negative sign is due to the fact that the mass is falling downwards.
Replacing in (1) we get
<em>-490.5 = 50x'(t) ===> x'(t) = -9.81 m/sec
</em>
and
<em>x'(t) = -9.81 m/sec
</em>
This is a simple differential equation which can be solved directly by integration
<em>x(t) = -9.81*t + C </em>
where C is a constant, but we know that x(0) = 3000 m, so
<em>C = 3000
</em>
We then have our final equation of motion which is
x(t) = -9.81*t + 3000
b) When will the object strike the ground?
The object will strike the ground for the instant t for which x(t) = 0
<em>x(t) = 0 ===> 9.81*t = 3000 ===> t = 3000/9.81 ===> </em><em>t = 305.81 sec
</em>
