ME 012 Engineering Dynamics
Lecture 2
Rectilinear Kinematics: Erratic Motion (Chapter 12, Section3)
Thursday,
Jan. 17, 2013
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
12.3 Rectilinear Kinematics: Erratic Motion
Today’s Objectives:
Determine position, velocity, and
acceleration of a particle using graphs.
In-Class Activities:
Applications
s-t, v-t, a-t, v-s, and a-s diagrams
Group Problem Solving
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
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12.3 Rectilinear Kinematics: Erratic Motion
Graphing provides a good way to handle complex
motions that would be difficult to describe with
formulas. Graphs also provide a visual description of
motion and reinforce the calculus concepts of
differentiation and integration as used in dynamics.
The approach builds on the facts that slope and
differentiation are linked and that integration can be
thought of as finding the area under a curve.
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
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12.3 Rectilinear Kinematics: Erratic Motion
s-t GRAPH
Plots of position vs. time can be used to find
velocity vs. time curves. Finding the slope of
the line tangent to the motion curve at any
point is the velocity at that point
(instantaneous velocity or v = ds/dt).
s-t to v-t
Therefore, the v-t graph can be constructed
by finding the slope at various points along
the s-t graph.
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
4
12.3 Rectilinear Kinematics: Erratic Motion
v-t GRAPH
Plots of velocity vs. time can be used to find
acceleration vs. time curves. Finding the slope of
the line tangent to the velocity curve at any point is
the acceleration at that point (instantaneous
acceleration or a = dv/dt).
v-t to a-t
Also, the distance moved (displacement) of the
particle is the area under the v-t graph during time
∆t.
Therefore, the a-t graph can be constructed by
finding the slope at various points along the v-t
graph.
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
5
12.3 Rectilinear Kinematics: Erratic Motion
v-t GRAPH
Given the v-t curve, we can also calculate
the change in position (∆s) during a time
period… this is the area under the v-t curve
by the definition of an integral:
v-t to s-t
∆ =
d
Area under
v-t graph
So we can construct an s-t graph from an v-t
graph if we know the initial position of the
particle.
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
6
12.3 Rectilinear Kinematics: Erratic Motion
a-t GRAPH
Given the a-t curve, the change in velocity
(∆v) during a time period is the area under
the a-t curve.
a-t to v-t
So we can construct a v-t graph from an a-t
graph if we know the initial velocity of the
particle.
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
7
12.3 Rectilinear Kinematics: Erratic Motion
a-s GRAPH
A more complex case is presented by the a-s graph.
The area under the acceleration versus position
curve represents the change in velocity
(recall ∫ a ds = ∫ v dv ).
1
2
−
=
d
a-s to v-s
1
2
−
=
d
Area under
a-s graph
This equation can be solved for v1, allowing you
to solve for the velocity at a point. By doing this
repeatedly, you can create a plot of velocity
versus distance.
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
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12.3 Rectilinear Kinematics: Erratic Motion
v-s GRAPH
v-s to a-s
Another more complex case is presented by
the v-s graph. By reading the velocity v at a
point on the curve and multiplying it by the
slope of the curve (dv/ds) at this same point,
we can obtain the acceleration at that point:
a = v (dv/ds)
Thus, we can obtain a plot of a vs. s from the vs curve.
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
9
12.3 Rectilinear Kinematics: Erratic Motion
EXAMPLE 1 (in text)
In many experiments, a velocity versus position (v-s)
profile is obtained.
If we have a v-s graph for the motorcycle, can we
construct the a-s graph of the motion and determine
the time needed for the motorcycle to reach the
position s = 400 feet?
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
10
12.3 Rectilinear Kinematics: Erratic Motion
EXAMPLE 1: Solution
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
11
12.3 Rectilinear Kinematics: Erratic Motion
EXAMPLE 2
Given: v-t graph for a train moving between two stations
Find:
a-t graph and s-t graph over this time interval
Think about your plan of attack for the problem!
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
12
12.3 Rectilinear Kinematics: Erratic Motion
EXAMPLE 2: Solution
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
13
12.3 Rectilinear Kinematics: Erratic Motion
EXAMPLE 3
Given:
The v-t graph shown. The car
starts from rest at s = 0
Find:
1) The a-t graph and determine
the maximum acceleration
during the 30 s interval.
2) Draw the s-t graph and
determine the average speed
and the distance travelled for
the 30 s interval.
Plan:
Find slopes of the curves and draw the v-t graph. Find the area under the
curve… that is the distance traveled. Finally, calculate equation for s-t
diagram and find average speed (using basic definitions!).
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
14
12.3 Rectilinear Kinematics: Erratic Motion
EXAMPLE 4: Solution (1/2)
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
15
12.3 Rectilinear Kinematics: Erratic Motion
EXAMPLE 4: Solution (2/2)
ME 012 Engineering Dynamics: Lecture 2
J. M. Meyers, Ph.D. (jmmeyers@uvm.edu)
16