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B Velocity after Free Fall and after Motion down an Incline

  1. Jul 12, 2018 #1
    I was doing some conceptual problems quickly before moving on to the numerical stuff when I glanced at this answer given in the solutions manual, and it surprised me:

    Which object has greater speed at the bottom of its fall, object A of mass m dropped freely from height h or object B of mass m moving down an incline (no friction) from height h?

    The manual says A. I googled around and found different people saying different things on this! I thought the answer was just that they have the same speed.

    The manual gave this as its explanation: for A, v = sqrt(2gh), whereas for B, v = sqrt(2g(sinθ)h). Therefore, A is greater.

    One of the online answers gave this: both A and B start with PE = mgh; so, at the end, all PE is now KE; so, KE = mgh = 1/2(mv^2); so both A and B have the same v, namely v = sqrt(2gh).

    My analysis is that the manual mistakenly gives g(sinθ) for the acceleration on the incline. Maybe they were thinking of the force component?

    I’d be grateful for help getting clear on this.
     
  2. jcsd
  3. Jul 12, 2018 #2
    You are correct, and the manuel is incorrect.
     
  4. Jul 12, 2018 #3

    jbriggs444

    User Avatar
    Science Advisor

    That formula would be accurate for a ramp of [diagonal] length h, not for a ramp of [vertical] height h.
     
  5. Jul 12, 2018 #4

    Dale

    Staff: Mentor

    Yes, as you can see from a simple application of the conservation of energy. In both cases the same amount of gravitational PE was lost, in both cases there were no friction losses, so all the PE goes into KE meaning they have the same KE and therefore the same speed.
     
  6. Jul 12, 2018 #5
    Thanks for the clarifications! I can see it more clearly now.
     
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