ES 102
Dynamics of Rigid Bodies
Problem Set No. 2
KINEMATICS OF PARTICLES
General Instructions: All solutions must be handwritten. The format of the Problem Set is
presented in the last part of this file.
1. A snowboarder starts from rest at the top of a double black diamond hill. As she rides
down the slope, GPS coordinates are used to determine her displacement as a
function of time: 𝒙 = 𝟎. 𝟓𝒕𝟑 + 𝒕𝟐 + 𝟐𝒕 where x and t are expressed in feet and seconds,
respectively. Determine the position, velocity, and acceleration of the boarder when
𝒕 = 𝟓 seconds.
2. The motion of a particle is defined by the relation 𝒙 = 𝟐𝒕𝟑 − 𝟗𝒕𝟐 + 𝟏𝟐𝒕 + 𝟏𝟎, where x and
t are expressed in feet and seconds, respectively. Determine the time, the position, and
the acceleration of the particle when 𝒗 = 𝟎.
3. The vertical motion of mass A is defined by the relation 𝒙 = 𝟏𝟎 𝐬𝐢𝐧 𝟐𝒕 +
𝟏𝟓 𝐜𝐨𝐬 𝟐𝒕 + 𝟏𝟎𝟎, where x and t are expressed in millimeters and seconds,
respectively. Determine (a) the position, velocity, and acceleration of
A when 𝒕 = 𝟏 s, (b) the maximum velocity and acceleration of A.
4. The acceleration of a particle is defined by the relation 𝒂 = 𝟑𝒆−𝟎.𝟐𝒕 ,
where a and t are expressed in ft/s2 and seconds, respectively. Knowing that x = 0 and
v = 0 at t = 0, determine the velocity and position of the particle when t = 0.5 s.
5. The acceleration of a particle is defined by the relation 𝒂 = −𝒌𝒗𝟐.𝟓 , where k is a
constant. The particle starts at x = 0 with a velocity of 16 mm/s, and when x = 6 mm,
the velocity is observed to be 4 mm/s. Determine (a) the velocity of the particle when
x = 5 mm, (b) the time at which the velocity of the particle is 9 mm/s.
6. An airplane begins its take-off run at A with zero velocity and a constant acceleration
a. Knowing that it becomes airborne 30 seconds later at B and that the distance AB is
900 m, determine
(a) The acceleration a
(b) The take- off velocity vB
7. A motorist is traveling at 54 km/hr when she observes that a traffic light 240 m ahead of
her turns red. The traffic light is timed to stay for 24 seconds. If the motorist wishes to
pass the light without stopping just as it turns green again, determine
(a) The required uniform deceleration of the car,
(b) The speed of the car as it passes the light.
8. Block A (Fig. P11.56) starts from rest at t = 0 and moves downward with
a constant acceleration of 6 in./s2. Knowing that block B moves up
with a constant velocity of 3 in./s, determine
(a) the time when the velocity of block C is zero,
(b) the corresponding position of block C.
9. Automobiles A and B are traveling in adjacent highway lanes and at t = 0 have the
positions and speeds shown. Knowing that automobile A has a constant acceleration of
1.8 ft/s2 and that B has a
constant deceleration of 1.2 ft/s2, determine
(a) when and where A will overtake B,
(b) the speed of each automobile at that time.
10. Slider block B moves to the right with a constant
velocity
of
300 mm/s. Determine
(a) the velocity of slider block A,
(b) the velocity of portion C of the cable,
(c) the velocity of portion D of the cable,
(d) the relative velocity of portion C of the cable with respect to slider block A.
11. A particle which moves in two-dimensional motion
has coordinates given in inches by
𝒙 = 𝒕𝟐 − 𝟒𝒕 +
𝟐𝟎 and 𝒚 = 𝟑𝒔𝒊𝒏𝟐𝒕, where the time t is in seconds.
Determine the magnitudes of the velocity v and the
acceleration a and the angle θ between these two
vectors at time t= 3 sec.
12. An airplane used to drop water on brushfires is flying horizontally in a straight line at 180
mi/h at an altitude of 300 ft. Determine the distance d at which the pilot should release
the water so that it will hit the fire at B.
13. A sports car is to travel from A to B in exactly 40 seconds. The distance between A to B is
540 m. The car accelerates at 1.8 m/sec2, then travel at a constant speed for some time
before it decelerates at 1.8 m/sec2. The car starts from rest at A and comes to rest at B.
Find the maximum speed. Draw the a-t, v-t and s-t graphs to solve the problem.
14. In
slow-pitch
softball,
the
underhand pitch must reach a
maximum height of between 1.8 m
and 3.7 m above the ground. A
pitch is made with an initial velocity
vo with a magnitude of 13 m/s at
an angle of 33° with the horizontal.
Determine (a) if the pitch meets
the maximum height requirement, (b) the height of the ball as it reaches the batter.
15. The basketball player likes to release his foul shots at an angle θ=
50° to the horizontal as shown. What initial speed vo will cause the
ball to pass through the center of the rim?
16. A tennis player serves the ball
at a height h= 2.5m with an
initial velocity of vo at an angle
of 5° with the horizontal.
Determine the range of vo for
which the ball will land in the
service area that extends to
6.4m beyond the net.
17. A train travels between two stations 792 meters apart in a minimum time of 41 seconds. The
train accelerates at 2.4 m/sec2, runs at a fast speed for some time, and decelerates at 2.4
m/sec2 until it comes to rest at the second station. If the train starts from rest at the first
station and comes to a stop at the second station, what is the maximum speed? How long
does it travel at this top speed? Draw the a-t, v-t and s-t graph to answer the problem.
18. A body starting from point E is given a uniformly acceleration from zero at E to 1.8 m/s^2
at F in 4 sec. The initial velocity at E is 3 m/s. Find the distance traveled at the end of 4
seconds. Solve this problem using motion curves.
19. A car starts from rest and reaches a speed of 48 m/s in 15 sec. The acceleration increases
from zero uniformly with time for the first 6 seconds after which it remains constant.
Determine the constant acceleration after 6 seconds. Also, determine the velocity after 6
seconds. Finally, determine the distance traveled in 15 seconds. Draw the a-t, v-t, and x-t
graphs then solve the problem.
20. A particle moving along the positive x-direction with an initial velocity of 12 m/s is subjected
to a retarding force that gives it a negative acceleration which varies linearly with time for
the first 4 seconds. For the next 5 seconds, the force is constant and the acceleration
remains constant. Compute the velocity at 𝑡 = 4 𝑠𝑒𝑐𝑜𝑛𝑑𝑠. Compute the velocity at 𝑡 =
9 𝑠𝑒𝑐𝑜𝑛𝑑𝑠. Compute the distance traveled by the particle from its position at 𝑡 = 0 to the
point where it reverses its position. Solve this problem using motion curves.
Format of PROBLEM SET
Paper Size: LEGAL (8.5 X 13)
Note: Cover page and borders can be computerized.
Color Coding:
Purple- Section 2A
Orange- Section 2B
Pink- Section 2C