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Physics 1D problems: Please help !

  1. Sep 12, 2004 #1
    Problem 1

    Ball A is dropped from the top of a building of height H at the same instant ball B is thrown vertically upward from the ground. First consider the situation where the balls are moving in opposite directions when the collide. If the speed of ball A is m times the speed of ball B when they collide, find the height at which they collide in terms of H and m. Take x = 0 at the ground, positive upward.

    1.1 With two equations, describe the conditions at the collision (position and velocities of the balls).

    - I got Xa = H - 1/2gt2
    Xb = Vob(t) - 1/2gt2

    At collision: Xa = Xb
    H = Vob(t)

    Va = -gt
    Vb = Vob - gt

    At collision, Va = Vb

    t = mVob/2g

    so Va = -g(mVob)/2g
    Vb = Vob - g(mVob)/2g

    1.2 Write the expressions for position and velocity of the balls as a function of time.

    See above

    1.3 Solve the above equations to find the height at which the balls collide. Your answer should be expressed as a fraction of the height of the building H and it should depend on the speed ratio m.

    I tried solving it but couldn't eliminate the Vob....help!

    1.4 Now suppose that m can be negative (i.e. balls A and B are moving in the same direction when they collide). Use the formula derived above to graph the height of the collision (again expressed as a fraction of the building height H) as a function of m for -5 < m < 5. Are there values of m for which the answer is unphysical?

    Problem 2

    Suppose, for a change, the acceleration of an object is a function of x, where a(x) = bx and b is a constant with a value of 2 seconds-2. In order to solve this problem you should use the chain rule: for arbitrary variables u, v and t, remember that dr/dt = (dr/ds) * (ds/dt).

    2.1 If the speed at x = 1 m is zero, what is the speed at x = 3 m? Be sure to show your work.

    By integration, should V = x squared?

    2.2 How long does it take to travel from x = 1 to x = 3 m?

    Problem 3

    A small rock sinking through water experiences an exponentially decreasing acceleration as a function of time, given by a(t) = ge-bt, where b is a positive constant that depends on the shape and size of the rock and the physical properties of water.

    3.1 Derive an expression for the position of the rock as a function of time, assuming the initial speed of the rock is zero.

    3.2 Show that the rock's acceleration can be written in a simple form involving its speed v: a = g - bv (still assuming that its initial speed is zero). This is, perhaps, a more common form of expressing acceleration in the presence of drag.
     
  2. jcsd
  3. Sep 12, 2004 #2

    Doc Al

    User Avatar

    Staff: Mentor

    comments on Prob 1

    Good.
    Good.

    No. Speed(a) = m x speed(b). Note: speed, not velocity. So that means:
    gt = m(v0b -gt)
     
  4. Sep 12, 2004 #3
    and...

    thanks doc, but I still don't see how I can eliminate Vob...
     
  5. Sep 12, 2004 #4

    Doc Al

    User Avatar

    Staff: Mentor

    Combine these two equations:
    (1) gt = m(v0b -gt)
    (2) H = V0b(t)

    Eliminate V0b, solve for t.
     
  6. Sep 28, 2005 #5
    mhmm if m=2... i got Xa = (2/3)H...

    is that right?
     
    Last edited: Sep 28, 2005
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