Finding time of travel by falling object in drag coeeficient

In summary, an object with mass 10kg is dropped from a height of 200m. Using the equation mv'(t) = −mg − kv(t) with a constant value of k = 2.5Nsm^-1, we can estimate the time it takes for the object to hit the ground. By solving the linear equation 0= -39t - 156.8e^(-1/4) + 156.8, we can determine the time to be approximately 3.9 seconds.
  • #1
vorse
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0

Homework Statement



An object with mass 10kg is dropped from a height of 200m. Given that the constant k in
the equation is 2.5Nsm^-1
mv'(t) = −mg − kv(t)

approximately how many seconds does the object hit the ground?



Homework Equations



v' + (k/m)v = -g



The Attempt at a Solution



p(t) = k/m h(t) = (k/m)t e^h(t) = e^(k/m)t

e^(k/m)t v = -gm/k e^(k/m)t +C

v(t) = -gm/k + Ce^-(k/m)t

v(0) = 0 ; C = gm/k

v(t) = -gm/k + gm/k e^-(k/m)t

S(t) - S(0) = integration of v(t) = -gmt/k - gm^2/k^2 e^-kt/m ; evaluate from 0 to t

0 = -39t - 156.8e^(-1/4) + 156.8 ; after substituting m = 10kg ; g = -9.8; and k = 2.5


Now I must solve for t, but how can I do this? I can't apply the quadratic formula can I?
 
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  • #2
Solve 0= -39t - 156.8e^(-1/4) + 156.8 for t? That's a simple linear equation.

39t= -156.8e^(-1/4) + 156.8 so t= (-156.8e^(-1/4) + 156.8)/39.
 
  • #3
0 = -39t - 156.8e^(-1/4)t + 156.8 ; after substituting m = 10kg ; g = -9.8; and k = 2.5

sorry, forgot the t from this equation; if you follow the work I did, you'll see. Still, not sure how to solve for t
 
  • #4
Bump*
Don't know the rule for bumping threads, but if this is illegal, I'll never do this again.
 

What is the equation used to find the time of travel for a falling object in drag coefficient?

The equation used is t = (m/ρACd) * (2V₀/g) * (1 - e^(-gρAt/m)) where t is the time of travel, m is the mass of the object, ρ is the density of the fluid, A is the cross-sectional area of the object, Cd is the drag coefficient, V₀ is the initial velocity of the object, and g is the acceleration due to gravity.

How does the density of the fluid affect the time of travel for a falling object in drag coefficient?

The greater the density of the fluid, the greater the drag force on the object, which results in a shorter time of travel. Conversely, a lower density fluid will result in a longer time of travel.

What is the significance of the drag coefficient in calculating the time of travel for a falling object?

The drag coefficient is a measure of how much resistance an object experiences while moving through a fluid. It is a crucial factor in calculating the time of travel for a falling object as it directly affects the drag force, and therefore, the acceleration and velocity of the object.

What are some common assumptions made when using the equation to find the time of travel for a falling object in drag coefficient?

Some common assumptions made include: constant drag coefficient throughout the fall, no change in the object's shape or orientation during the fall, and negligible effects from air density and gravity on the object's motion.

How accurate is the equation for finding the time of travel for a falling object in drag coefficient?

The accuracy of the equation depends on the accuracy of the input values and the validity of the assumptions made. In real-world scenarios, there may be other factors at play that can affect the object's motion, such as wind speed, turbulence, and air density variations. Therefore, the equation may provide an estimate rather than an exact value for the time of travel.

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