Respuesta :
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
