modeling w/ first order: 4 kg mass projected vertically upward w/ v_0 = 31 m/sec

[english102]

Good afternoon,

I am working on a math problem that i need some help with. I am just stuck on it and can not figure it out. I have worked on this problem multiple times, but i keep getting it wrong. Any help will be great. Thank you.


A body of mass 4 kg is projected vertically upward with an initial velocity 31 meters per second.

We assume that the forces acting on the body are the force of gravity and a retarding force of air resistance with direction opposite to the direction of motion and with magnitude [FONT=MathJax_Math-italic]c[FONT=MathJax_Main]|[/FONT][FONT=MathJax_Math-italic]v[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]t[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]|[/FONT][/FONT] where [FONT=MathJax_Math-italic]c[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]0.45[/FONT][FONT=MathJax_Math-italic]k[/FONT][FONT=MathJax_Math-italic]g[/FONT][FONT=MathJax_Math-italic]s[/FONT]c[/FONT] and [FONT=MathJax_Math-italic]v[FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]t[/FONT][FONT=MathJax_Main])[/FONT][/FONT] is the velocity of the ball at time t. The gravitational constant is [FONT=MathJax_Math-italic]g[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]9.8[/FONT][FONT=MathJax_Math-italic]m[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Math-italic]s[/FONT][FONT=MathJax_Main]2[/FONT]g

a) Find a differential equation for the velocity
v:

dv/dt=???


[/FONT]
b) Solve the differential equation in part a) and find a formula for the velocity at any time [FONT=MathJax_Math-italic]tt[/FONT]:

v(t)=????


C) Find a formula for the position function at any time
t, if the initial position is s(0)=0:



s(t)=????

 

[Deleted member 4993]

Good afternoon,

I am working on a math problem that i need some help with. I am just stuck on it and can not figure it out. I have worked on this problem multiple times, but i keep getting it wrong. Any help will be great. Thank you.


A body of mass 4 kg is projected vertically upward with an initial velocity 31 meters per second.

We assume that the forces acting on the body are the force of gravity and a retarding force of air resistance with direction opposite to the direction of motion and with magnitude [FONT=MathJax_Math-italic]c[FONT=MathJax_Main]|[/FONT][FONT=MathJax_Math-italic]v[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]t[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]|[/FONT][/FONT] where [FONT=MathJax_Math-italic]c[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]0.45[/FONT][FONT=MathJax_Math-italic]k[/FONT][FONT=MathJax_Math-italic]g[/FONT][FONT=MathJax_Math-italic]s[/FONT]c[/FONT] and [FONT=MathJax_Math-italic]v[FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]t[/FONT][FONT=MathJax_Main])[/FONT][/FONT] is the velocity of the ball at time t. The gravitational constant is [FONT=MathJax_Math-italic]g[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]9.8[/FONT][FONT=MathJax_Math-italic]m[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Math-italic]s[/FONT][FONT=MathJax_Main]2[/FONT]g

a) Find a differential equation for the velocity v:

dv/dt=???[/FONT]
b) Solve the differential equation in part a) and find a formula for the velocity at any time [FONT=MathJax_Math-italic]tt[/FONT]:

v(t)=????

C) Find a formula for the position function at any time t, if the initial position is s(0)=0: s(t)=????

Since you have worked on this problem multiple times, please share some of those work with us (that would tell us where do we start to help you.)

 

[HallsofIvy]

I am working on a math problem that i need some help with. I am just stuck on it and can not figure it out. I have worked on this problem multiple times, but i keep getting it wrong. Any help will be great. Thank you.

A body of mass 4 kg is projected vertically upward with an initial velocity 31 meters per second.

We assume that the forces acting on the body are the force of gravity and a retarding force of air resistance with direction opposite to the direction of motion and with magnitude c|v(t)| where c=0.45kgsc and v(t) is the velocity of the ball at time t. The gravitational constant is g=9.8m/s2g

a) Find a differential equation for the velocity
v:

dv/dt=???

"Force equals mass times acceleration" so "acceleration", which is dv/dt, is equal to the force, divided by the mass. Your are told that the mass is 4 kg. One part of the force is that of gravity which is mg so the acceleration is the constant g. Assuming your coordinate system is set up with positive upward, that is -g= -9.8. Another part of the force is the retarding force 0.45|v|. divided by mass= 4 that gives an addition to acceleration of 0.1125|v

dv/dt= -9.8- 0.1125|v|.

b) Solve the differential equation in part a) and find a formula for the velocity at any time tt:

v(t)=????

To solve the equation dv/dt= -9.8- 0.1125|v| you will have to look at two possible situations. If v is positive (the object is going upward) then the equation is dv/dt= -9.8- 0.1125v. You can write that as dv/(9.8+ 0.1125v)= -dt and integrate. For what value of t does v become 0 and then negative?

If v is negative (the object is coming back down) then the equation is dv/dt= -9.8+ 0.1125v. You can write that as dv/(0.1125v- 9.8)= dt and integrate.

C) Find a formula for the position function at any time t, if the initial position is s(0)=0:

s(t)=????

v= dx/dt so dx= vdt. Integrate the velocity function with respect to time.