Calculus Hw?

a stone was thrown upward from the surface of Titan (the largest moon of the planet saturn) with velocity 8 meters/sec. find the maximum height (round to two decimal places) of the stone given that the acceleration due to gravity on the surface of Titan is 235.1 meters/sec^2.

Show steps please I am very confused!

3 Answers

Relevance
  • 6 months ago

    Let the height of the stone at time t be y m from the surface

    of Titan, then

    y"=-235.1

    =>

    y'd(y')/dy=-235.1

    =>

    [(y')^2]/2=-235.1y+C

    y=0, y'=8

    =>

    32=C

    =>

    [(y')^2]/2=32-235.1y

    When the stone reaches the

    max. height, y'=0

    =>

    max.y=32/235.1=0.1361 m/s

    approximately.

    • Login to reply the answers
  • 6 months ago

    Is this really the problem statement? 235.1 m/s^2 is nearly 24 times Earth's surface gravity. No planet has a gravitational acceleration anywhere near that. The strongest acceleration (probably well below the surface) is a little over one-tenth of that.

    Anyway, to solve height in terms of velocity, equate the kinetic energy to the work that energy need to do to raise a mass m by a height h with a gravitational acceleration g:

    (1/2)mv^2 = mgh . . . . energy = work

    v^2 = 2gh . . . . divide by m to cancel, multiply by 2 to get rid of the fraction

    h = v^2 / (2g)

    That's it. Plug numbers to get h = 8^2 / (2 * 235.1) = 64 / 470.2 = 0.136 m. Much simpler than solving for elapsed time first.

    • Login to reply the answers
  • a(t) = -235.1

    v(t) = -235.1 * t + C

    v(0) = 8

    8 = -235.1 * 0 + C

    8 = C

    v(t) = -235.1 * t + 8

    s(t) = -117.55 * t^2 + 8t + C

    s(0) = 0

    s(t) = -117.55 * t^2 + 8t + 0

    s(t) = 8t - 117.55 * t^2

    v(t) = 0. This tells you when we're at a maximum height.

    v(t) = -235.1 * t + 8

    0 = 8 - 235.1 * t

    0 = 80 - 2351 * t

    2351 * t = 80

    t = 80 / 2351

    Plug that value for t in to s(t)

    s(t) = 8t - 117.55 * t^2

    s(80/2351) = 8 * (80/2351) - 117.55 * (80/2351)^2

    s(80/2351) = (80/2351)^2 * (8 * 2351/80 - 117.55)

    s(80/2351) = (80/2351)^2 * (235.1 - 117.55)

    s(80/2351) = (80/2351)^2 * (117.55)

    s(80/2351) = 0.13611229264142917907273500638026....

    It'll achieve a maximum height of 13.61 cm

    • Login to reply the answers
Still have questions? Get your answers by asking now.