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Assuming that only air resistance and gravity act on a falling object, we can find that the velocity of the object, v, must obey the differe
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Assuming that only air resistance and gravity act on a falling object, we can find that the velocity of the object, v, must obey the differential equation dv m mg bv dt . Here, m is the mass of the object, g is the acceleration due to gravity, and b > 0 is a constant. Consider an object that has a mass of 100 kilograms and an initial velocity of 10 m/sec (that is, v(0) = 10). If we take g to be 9.8 m/sec2 and b to be 5 kg/sec, find a formula for the velocity of the object at time t. Further, find the terminal (or limiting) velocity of the object. Circle your velocity formula and the terminal velocity.
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2021-09-04T12:16:04+00:00
2021-09-04T12:16:04+00:00 1 Answers
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Answer:
v = 196 – 186*e^( – 0.05*t )
v-terminal = 196 m/s
Explanation:
Given:
– The differential equation for falling object velocity v in gravity with air resistance is given by:
m*dv/dt = m*g – b*v
– The initial conditions and constants are as follows:
v(0) = 10 , m = 100 kg , b = 5 kg/s , g = 9.8 m/s^2
Find:
– Find a formula for the velocity of the object at time t. Further, find the terminal (or limiting) velocity of the object. Circle your velocity formula and the terminal velocity.
Solution:
– Rewrite the differential equation in te form:
dv/dt + (b/m)*v = g
– The integration factor function P(t) = b/m. The integrating factor u(t) is:
u(t) = e^∫P(t).dt
u(t) = e^∫(b/m).dt
u(t) = e^[(b/m).t]
– Solve the differential equation after expressing in form:
v.u(t) = ∫u(t).g.dt
v.e^[(b/m).t] = g*∫e^[(b/m).t].dt
v.e^[(b/m).t] = g*m*e^[(b/m).t] / b + C
v = g*m/b + C*e^[-(b/m).t]
– Apply the initial conditions v(0) = 10 m/s and evaluate C:
10 = 9.8*100/5 + C*e^[-(b/m).0]
10 = 9.8*100/5 + C
C = -186
– The final ODE solution is:
v = 196 – 186*e^( – 0.05*t )
– The Terminal velocity vt can be expressed by a limiting value for v(t), where t ->∞.
vt = Lim t ->∞ ( v(t) )
vt = Lim t ->∞ ( 196 – 186*e^( – 0.05*t ) )
vt = 196 – 0 = 196 m/s