Download Advanced Problems 1

Advanced Problems 1
These problems will contain:
1. Basic concepts of velocity and
acceleration.
2. Usage of proper physics 1
kinematics equations.
3. Some calculus techniques,
such as derivatives and
integration.
1. An indestructible bullet 2cm long is fired
straight through a board that is 10cm thick.
The bullet strikes the board with a speed of
420m/s and emerges with a speed of 280m/s.
–
What is the average acceleration of the bullet as it
passes through the board?
– What is the total time that the bullet is in
contact with the board?
– What thickness of board would it take to stop
the bullet (assuming the bullet’s acceleration
though the board remains the same)
2. A student throws a set of keys vertically
upward to her sorority sister, who is in a
window 4m above. The keys are caught 1.5s
later by the sisters outstretched hand.
– With what initial velocity were the keys
thrown?
• What was the velocity of the keys just before
they were caught?
3. The height of a helicopter above the ground is
given by h = 3t3, where h is in meters and t is
in seconds. After 2 seconds, the helicopter
releases a small mailbag. How long after its
release does the mailbag reach the ground?
4. Automotive engineers refer to the time rate of
change of acceleration as the “jerk”. If an
object moves in one dimension such that its
jerk J is constant,
a. Determine expressions for its acceleration,
velocity, and position given that its initial
acceleration, velocity and speed are ai, vi, and
xi respectively.
b. Show that
a  ai  2 J (v  vi )
2
2
5. The speed of a bullet as it travels down a
barrel of a rifle toward the opening is given by
v = -5x107t2 + 3x105t. The acceleration of the
bullet as it just leaves the barrel is zero.
a. Determine the acceleration and position of the
bullet as a function of time when the bullet is in
the barrel.
b. Determine the length of time the bullet is
accelerated.
c. Find the speed at which the bullet leaves the
barrel.
d. What is the length of the barrel?
6. A daring ranch hand sitting on a tree limb
wishes to drop vertically onto a horse galloping
under a tree. The speed of the horse is 10m/s,
and the distance from the limb to saddle is 3m.
What must be the horizontal distance between
the limb and the saddle when the ranch hand
drops?
7. A test rocket is fired vertically upward from a
well. A catapult gives it an initial velocity of
80m/s at ground level. Subsequently, its
engines fire and it accelerates upward at 4m/s2
until it reaches an altitude of 1000m. At that
point the engines fail, and the rocket goes into
free fall.
a. How long is the rocket in motion above the
ground?
b. What is its maximum altitude?
c. What is its velocity just before it hits the
ground?
8. A physics student and a mountain climber
climbs a 50m cliff that overhangs a calm pool of
water. He throws two stones vertically
downward, 1second apart, and observes that
they cause a single splash. The first stone has
an initial speed of 2m/s.
• How long after the release of the first stone do
the stones hit the water?
• What was the initial velocity of the second
stone?
• What is the velocity of each stone at the instant
the two hit the water?
9. You are standing on the ground at the origin of
a coordinate system. An airplane flies over you
with constant velocity parallel to the x axis and
at a constant height of 7.6x103m. At t=0, the
plane is directly above you. At t=30s, the
position vector from you to it is given by
P30=(8.04x103m)i + (7.6x103m)j. Determine the
magnitude and orientation of the plane’s
position at t=45s.
10.Given the displacement vectors
A= (3i-4j+4k)m and B= (2i+3j-7k)m find the
magnitudes of C = A + B and D = 2A – B also
expressing each in terms of its x, y, and z
components.
11.A radar station locates a sinking ship at range
17.3km and bearing 136° clockwise from north.
From the same station a rescue plane is at
horizontal range 19.6km 153° clockwise from
north. With elevation 2.2 km.
a. Write the vector displacement from plane to
ship letting i represent east, j represent north,
and k represent up.
b. How far apart are the plane and ship?
12.A vector is given by R=2i + 1j +3k.
a. Find the magnitudes of the x, y, and z
components.
b. Find the magnitude of R.
c. Find the angles between R and the x, y, and z
axes.
13. In general, the instantaneous position of an
object is specified by its position P leading from
a fixed origin to the location of the object.
Suppose that P= 4i+3j-2tj where P is in meters
and t is in seconds what is the derivative of this
position function? What does this derivative
represent about the object?
14. A jet airliner, moving initially at 300mi/h to
the east, suddenly enters a region where the
wind is blowing at 100mi/h in a direction 30°
north of east. What are the new speed and
direction of the aircraft relative to the ground?
15. A fish swimming in a horizontal plane has
velocity vi = (4i + 1j)m/s at a point in the ocean
whose displacement from a certain rock is ri =
(10i – 4j)m. After the fish swims with constant
acceleration for 20s, its velocity is v = (20i +
5j)m/s
a. What are the components of the acceleration?
b. What is the direction of the acceleration with
respect to the unit vector i?
c. Where is the fish at t=25s if it maintains its
original acceleration and in what direction is it
moving?
16. A particle initially located at the origin has an
acceleration of a = 3j m/s2 and an initial
velocity of vi = 5i m/s.
a. Find the vector position and velocity at any
time t.
b. Find the coordinates and speed of the particle
at t=2s.
17. A projectile is fired in such a way that its
horizontal range is equal to 3 times its
maximum height. What is the angle of
projection?
18. A placekicker must kick a football from a point
36m from the goal, and half the crowd hopes it
will clear the crossbar, which is 3.05m high.
When kicked, the ball leaves the ground with a
speed of 20m/s at an angle of 53° to the
horizontal.
a. By how much does the ball clear or fall short of
the crossbar?
b. Does the ball approach the crossbar while still
rising or while falling?
19. A train slows down as it rounds a sharp
horizontal curve, slowing from 90km/h to
50km/h in the 15 seconds that it takes to round
the curve. The radius of the curve is 150m.
Compute the acceleration at the moment the
train speed reaches 50km/h. Assume that the
train slows down at a uniform rate during the
15 second interval.
20. The determined coyote is out once more to capture the
roadrunner. The coyote wears Acme jet powered roller
skates which provide constant horizontal acceleration
of 15m/s2. The coyote starts off at rest 70m from the
edge of a cliff the instant that the roadrunner zips past
him in the direction of the cliff.
a.
If the roadrunner moves with constant speed,
determine the minimum speed he must have to reach
the cliff before the coyote.
b.
If the cliff is 100m above the floor of the canyon,
determine where the coyote lands after missing the
roadrunner (his skates still thrust while in midair).
c.
What is his impact velocity?