Math 2250
Lab 1
Due Sep 4, 2014
Name:
1. Verify that the exponential integral
Z
∞
y(x) = Ei(x) =
x
e−u
du
u
with x > 0, satisfies the differential equation:
d2 y 1 dy
= 0.
+
1
+
dx2
x dx
Useful identities:
Rb
a
f (u) du = −
Ra
b
f (u) du
d
dx
Rx
a
f (u) du = f (x).
The exponential integral and its differential equation are important in modeling water
flow to a pumping well through a porous ground medium.
Math 2250
Lab 1, Page 2 of 7
Due Sep 4, 2014
2. Resistance is a force acting opposite to the relative motion of an object. We now consider
that a feather of mass 0.0005 kg falls vertically from a very high place. When the feather
falls at a velocity v m/s, the air resistance acting against the feather is given by 0.02v 2
Newtons. Assume the gravity is 10 m/s2 .
(a) Let s represent the (downward) distance travelled by the feather and let t represent
time. Show that the motion of the feather is described by the following differential
equation:
2
d2 s
ds
− 10 = 0.
+ 40
2
dt
dt
Math 2250
Lab 1, Page 3 of 7
(b) Verify that the following function
1
1
s(t) = t +
ln(1 + e−40t ) + C,
2
40
where C is a constant, satisfies the differential equation in (a).
Due Sep 4, 2014
Math 2250
Lab 1, Page 4 of 7
Due Sep 4, 2014
3. (a) On the planet Gzyx, a ball dopped from a height of 20 ft hits the ground in 1 s. If
a ball is dropped from the top of a 200 ft tall building on Gzyx, how long will it
take to hit the ground? With what speed will it hit?
Math 2250
Lab 1, Page 5 of 7
Due Sep 4, 2014
(b) A person can throw a ball straight upward from the surface of the earth to a
maximum height of 144 ft. How high could this person throw the ball on the planet
Gzyx of part (a)?
Math 2250
Lab 1, Page 6 of 7
Due Sep 4, 2014
4. A car is traveling along a straight road. Let v(t) be the velocity function (in miles per
minute) described by
(
0.5t2
0≤t≤1
v(t) =
t − 0.5 1 < t ≤ 2
(a) Sketch the graph of the function v(t).
(b) Let x(t) be the position function of the curve with x(0) = 0. Find a formula for
x(t) for 0 ≤ t ≤ 2.
Math 2250
Lab 1, Page 7 of 7
(c) Find the acceleration of the car after it travels 0.1 mile.
Due Sep 4, 2014