**UIUC Physics 211 James Scholars Problem Set 1**

In this assignment we will do some problems in 1-d motion in a bit more depth.

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).

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

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*.

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 r, s 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.

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.