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Download Question A particle is projected vertically upward in a constant
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Transcript
Question A particle is projected vertically upward in a constant gravitational field with an initial speed of v0 . Show that if there is a retarding force proportional to the square of the speed, the speed of the particle when it returns to its initial position is v0 vT q , v02 + vT2 where vT is the terminal speed. Answer Do this question in 2 parts. Upward motion y 6 • ? mg ? mk ẏ 2 Using Newton’s 2nd law gives: mÿ = −mk ẏ 2 − mg dv dv dv = −kv 2 − g Put =v dt dt dy Z v dv = −y + C kv 2 + Y à ! 1 kv02 + g whence y = ln , 2k kv 2 + g where v0 is the initial upwards velocity. At the highest point v =Ã0. ! k02 + g 1 Height attained = ln 2k g Downward motion mk ẏ 2 6 • ? mg Using Newton’s 2nd law gives: 1 ÿ = k ẏ 2 − g à à ! ! g − kv 2 kv02 + g 1 1 ln ln y = + , (∗) 2k g 2k g where v = 0 at the highest r point. g 2 v Solving for v gives v = v2k+0g 0 k Terminal velocity occurs when there is nor net force on the mass while the g . particle is falling i.e. mkvt2 = mg ⇒ vt = k Therefore v = q 2 v0 vt v02 + vt2
