Review  Two Dimensional Motion



1.

An object experiences displacements s_{1}
ands_{2}, as shown in Fig. 1a above. Is
the resultant displacement s shown in Fig. 1b or Fig.
1c? Explain your answer.


2.

Show how two displacement vectors, one
of magnitude 6 m and the other of magnitude 8 m, can be combined
to give a resultant of (a) 2 m, (b) 14 m, and
(c) 10 m.


3.

Vector A has magnitude 10 cm and
makes an angle of 37^{o} with the xaxis. Vector B
has components B_{x} = 8 cm and B_{y} = 2
cm. Sketch the vector
C = A + 1/2 B and determine its components,
its magnitude, and the angle it makes with the Xaxis.


4.

A particle moves 7.00 m along the positive
Yaxis and then 7.07 m at angle of 45^{0} with the
Xaxis, as shown in Fig. 2 below. Find the resultant displacement
r = r_{1} + r_{2} of
the particle. Express r in terms of (a) unit vectors
i and j and (b) its magnitude and direction.



5.

A
pitcher throws a baseball with a horizontal velocity of 132
ft/s (90 mph) toward home plate that is
approximately 60 ft away. What is the vertical drop of the
ball when it reaches the plate? g = 32 ft/s^{2}.


6.

A young man practicing his pitching
stands on a cliff and tosses a rock at a velocity of 24 m/s
in a horizontal direction. At this instant a person in a car
at rest in a service station 90 m from the cliff "guns"
the car at a constant acceleration of 4.0 m/s^{2}
toward the cliff. The rock hits the car. (a) How long is the
rock in the air? (b) How high is the cliff?


7.

A ball rolls off the edge of a 1.0
mhigh table with a speed of 4.0 m/s. How far horizontally
from the edge of the table does the ball strike the floor?


8.

When an object is given an initial velocity
v_{ox} along a horizontal frictionless surface, it travels
a distance x = v_{ox}t in time t.
When an object is given an initial velocity vertically upward
of v_{oy} from an initial height y_{o}, it rises
until its velocity v_{y }= 0.
The maximum height y_{max }it rises can be found from
0 = v_{y}^{2} = v_{oy}^{2
}+ 2a_{y}(y_{max } y_{o}),
where a_{y }= 9.8 m/s^{2} » 10 m/s^{2}.
The time t it is in the air is found when
y = 0 = y_{o} + v_{oy}t + 1/2
a_{y}t^{2}.
The velocity at this time t is given by
v_{y}(t) = v_{oy }+ a_{y}t.
The principle of superposition states that when a body is
subjected to two or more separate influences it responds to
each without altering its response to the other. If the object
is given an initial velocity of v_{o} at angle Θ
such that the components of its velocity are the same v_{ox}
and v_{oy }as given above, what happens to the quantities
t, y_{max}, v_{y}(t), and x?


9.

Find and draw the horizontal and vertical
components of the projectile's velocity in Fig. 3 (a) below
at the initial position i, (b) at the highest point h, and
(c) just before it hits the ground at f. (d) Draw the
velocity vector at h and f. (e) Draw the acceleration
vectors for x = 20.0, 30.0 and 40.0 m.



10.

An object is shot into the air with an
initial velocity v_{o} = 25 m/s at an angle of 37^{0}
above the horizontal at a height of 20 m above the earth's
surface (Fig. 4 below). Take g = 10 m/s^{2}.
Find and draw the horizontal and vertical components
of the projectile's velocity in Fig. 4 (a) at the initial
position, (b) at the highest point. Find (c) the maximum height
y_{max} to which the object rises, (d) the time for
it to return to the earth, (e) the distance moved horizontally,
and (f) draw the horizontal and vertical components of the
velocity just before it hits the ground. (g) Draw the
velocity vector just before it hits the ground and find its
magnitude and direction.



11.

A boy throws a ball into the air as
hard as he can and then runs as fast as he can under the ball
in order to catch it. If his maximum speed in throwing the
ball is 20 m/s and his best time for a 20m dash is 3.0 s,
how high does the ball rise. Take g = 10 m/s^{2}.


12.

A driven golf ball just clears the top
of a tree that is 15 m high and is 30 m from the tee, and
then lands (with no roll or bounce) on the green, 180 m from
the tee. What was the initial velocity imparted to the golf
ball? Take g = 10 m/s^{2}.


13.

A small object of mass m rotates counterclockwise
on a horizontal frictionless plane at the end of a string
of length r = 0.4 m with constant speed v = 2.0
m/s. On Fig. 6 draw the magnitude and direction of (a) the
velocity and (b) the centripetal acceleration at points A,
B, and C. Are (c) the velocity and (d) the acceleration constant?
(d) Could you use this method to find how many “g’s”
this object could withstand. Explain your answers.



14.

(a) The period T of an object moving
with uniform circular motion is defined as the time for one
complete rotation. Find the period for the particle in #13.
(b) The frequency f of an object is defined as the number
of rotations it makes per second. Find the frequency of the
particle in #13. How is the frequency related to the period?
Write an expression for the centripetal acceleration in terms
of (c) π, r, and T, (d)
π, r, and
f.

