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Chapter 3: Q5E (page 115)

An object of mass 5 kg is given an initial downward velocity of 50 m/sec and then allowed to fall under the influence of gravity. Assume that the force in newtons due to air resistance is -10v, where v is the velocity of the object in m/sec. Determine the equation of motion of the object. If the object is initially 500 m above the ground, determine when the object will strike the ground.

Short Answer

Expert verified
  • The equation of motion of the object isx(t)=4.91t+22.55(1-e-2t)
  • The time takes the object hits the ground 97.24 sec.

Step by step solution

01

Find the velocity

ma = W - bvtma = mg - bvt

Velocity,

v(t)=mgb+(v-mgb)e-btmv(t)=5(9.81)10+(50-5(9.81)10)e-10t5v(t)=4.905+45.095e-2t

02

Find the equation of motion

x(t)=mgtb+mb(v-mgb)(1-e-btm)x(t)=4.905t+510(45.095)(1-e-2t)=4.91t+22.55(1-e-2t)

Hence the equation of motion is x(t)=4.91t+22.55(1-e-2t)

03

Find the value of t

Put x=500 and neglecting the exponential part

500=4.91t+22.55t=97.24sec

Hence, the time takes the object hits the ground 97.24 sec.

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