Find a formula for the velocity at any time ( in terms of k ): v(t) = (Ce-((k/4)(t)) -39.2)/k Find the limit of this velocity for a fixed time to as the air resistance coefficient k goes to 0. (Write to as to in your answer.) v(to) = -9.8t+9
Gravitational force
In nature, every object is attracted by every other object. This phenomenon is called gravity. The force associated with gravity is called gravitational force. The gravitational force is the weakest force that exists in nature. The gravitational force is always attractive.
Acceleration Due to Gravity
In fundamental physics, gravity or gravitational force is the universal attractive force acting between all the matters that exist or exhibit. It is the weakest known force. Therefore no internal changes in an object occurs due to this force. On the other hand, it has control over the trajectories of bodies in the solar system and in the universe due to its vast scope and universal action. The free fall of objects on Earth and the motions of celestial bodies, according to Newton, are both determined by the same force. It was Newton who put forward that the moon is held by a strong attractive force exerted by the Earth which makes it revolve in a straight line. He was sure that this force is similar to the downward force which Earth exerts on all the objects on it.
![A body of mass 4 kg is projected vertically upward with an initial velocity 9
meters per second.
The gravitational constant is g = 9.8m/s². The air resistance is equal to k|v| where
k is a constant.
Find a formula for the velocity at any time (in terms of k ):
v(t) = (Ce-((k/4)(t)) -39.2)/k
Find the limit of this velocity for a fixed time to as the air resistance coefficient k
goes to 0. (Write to as to in your answer.)
v(to) = -9.8t+9
How does this compare with the solution to the equation for velocity when there is
no air resistance?
This illustrates an important fact, related to the fundamental theorem of ODE and
called 'continuous dependence' on parameters and initial conditions. What this
means is that, for a fixed time, changing the initial conditions slightly, or changing
the parameters slightly, only slightly changes the value at time t.
The fact that the terminal time t under consideration is a fixed, finite number is
important. If you consider 'infinite' t, or the 'final' result you may get very different
answers. Consider for example a solution to y'= y, whose initial condition is
essentially zero, but which might vary a bit positive or negative. If the initial
condition is positive the "final" result is plus infinity, but if the initial condition is
negative the final condition is negative infinity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff479df57-fb9b-492e-9a82-54eb547f68e8%2F02299f4e-d447-4791-96e7-ddcb794f2126%2Fj3plykc_processed.png&w=3840&q=75)
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