1 Velocity and Acceleration
At the end of this module you should be able to:
a.
Solve problems using
s=
b.
c.
d.
1
(u + v )t
2
v = u + at
v 2 = u 2 + 2as
s = ut +
1 2
at
2
Explain what s, u, v, a and t represent.
Define angular velocity and angular acceleration, and solve problems.
Relate linear and angular acceleration and velocity, and solve problems.
Table of contents
1
1.1
1.2
1.2.1
1.2.2
1.2.3
1.2.4
1.3
1.3.1
1.3.2
1.4
1.4.1
1.4.2
1.4.3
1.5
Velocity and Acceleration ....................................................................................................................................1
Scalars and Vectors ..............................................................................................................................................2
Linear Motion.......................................................................................................................................................2
Linear displacement and distance ...............................................................................................................2
Linear velocity and speed ...........................................................................................................................2
Linear acceleration......................................................................................................................................3
Equations for linear uniformly accelerated motion.....................................................................................4
Angular Motion ..................................................................................................................................................10
Angular speed and velocity.......................................................................................................................10
Angular acceleration .................................................................................................................................10
Relationship between linear and angular motion................................................................................................11
Relationship between linear speed and angular speed ..............................................................................11
Relationship between angular speed and frequency of rotation ................................................................11
Relationship between linear acceleration and angular acceleration ..........................................................12
Uniform circular motion - (acceleration)............................................................................................................13
Page numbers on the same topic in , Applied Mechanics, 3rd Edition, Hannah & Hillier
Section in these notes
Section 1.1 - Section 1.2.4
Section 1.3 - Section 1.3.1
Section 1.3.2
Section 1.4 - 1.4.3
Section 1.5
Section in Hannah & Hillier
4.1 - 4.7
4.9
9.1 - 9.2
Not in the book
6.1
Fundamentals of Mechanics – Kinetics: Section 1 – Velocity and Acceleration
Page No. in Hanna & Hiller
64 - 6
73
154 - 156
Not in the book
102 - 104
1
1.1
Scalars and Vectors
A scalar quantity is one which can be completely specified by a number that expresses its magnitude in an appropriate
unit.
Some examples are:
Scalar quantity SI unit
Mass
kg
Length
m
Area
m2
A vector quantity requires a number (magnitude) and direction.
Some examples are:
Vector quantity SI unit
Velocity
m/s
Acceleration
m/s2
Force
N (or kg m/s2)
1.2
1.2.1
Linear Motion
Linear displacement and distance
The linear displacement is the length moved in a given direction - it is a vector quantity.
The magnitude of the displacement is the distance - a scalar quantity.
1.2.2
Linear velocity and speed
The linear velocity is the rate of change of displacement with time. As displacement is a vector so velocity is a vector.
The magnitude of the velocity is speed. It is the
of change of distance with time - hence it is a scalar.
If a body moves with uniform velocity then it must move in a fixed direction with constant speed.
The average speed of a body is the total distance moved divide by the total time taken.
Fundamentals of Mechanics – Kinetics: Section 1 – Velocity and Acceleration
2
Distance - time curve
A graph plotted for distance (s) against time (t), might look like that in Figure 1.1:
C
B
distance s
A
Figure 1.1: Distance-Time Curve
time t
As speed is rate of change of distance with time, the slope, gradient, of the s/t curve is the speed.
Over the linear section OA of the curve the speed must be uniform.
Between A and B the gradient is becoming less and less, hence the body is slowing down.
At B the body is stopped (distance is not increasing) and remains at rest between B and C.
1.2.3
Linear acceleration
The linear acceleration of a body is the rate of change of linear velocity with time. It is a vector.
If acceleration is uniform the speed must be increasing b
t in equal time intervals.
Worked example 1.1
A car is travelling along a straight road at 13 m/s. It accelerates uniformly for 15 s until it is moving at 25 m/s
Solution
change in velocity dv
=
time taken
dt
25 − 13
=
15
= 0 .8 m / s 2
Acceleration =
Fundamentals of Mechanics – Kinetics: Section 1 – Velocity and Acceleration
3
Speed - Time curve
A graph of speed (v) of a body plotted against time (t) might be as shown by the graph in figure 1.2:
D
speed v
C
B
A
t1
dt
t2
time t
Figure 1.2: Speed-Time Curve
As acceleration is rate of change of speed (v) with time (t), the slope, gradient, of the v/t curve is the speed.
In Figure 1.2 the gradient between A and B is increasing - hence acceleration - is increasing, between B and C it is
constant, between C and D it is decreasing.
If in the small time interval dt, the speed is v. The distance covered in the time dt is
ds = v dt
ds
v=
dt
The total distance s travelled in the time interval between t1and t2 is the integral of this i.e.
s = ∫ ds = ∫ v dt
t1
t2
This integral is the same as the area under the curve.
Thus the distance travelled in any time interval is the area under the v/t curve between the start and end time.
1.2.4
Equations for linear uniformly accelerated motio
If a body that is moving in a straight line and started with initial speed u undergoes a uniform acceleration a for a time t
until its velocity is v, then the speed time curve would look like that in Figure 1.3:
Fundamentals of Mechanics – Kinetics: Section 1 – Velocity and Acceleration
4