ME 2030
Dynamics
Practice Problems
Autumn 2025
Assigned: Wednesday August 27 - 8:00 am
Due: N/A
Mathematical tools for dynamics: Vector Algebra, Differential and Integral Calculus
Instructions
1. These practice problems are NOT graded. This is intended as a refresher for mathematical tools (pre-requisites) that you may use in this course.
2. Feel free to refer to this practice set in order to refresh mathematical techniques as
needed as the course progresses.
Problems
1. Vector operations: addition and subtraction
(a) A pedestrain moves 6 km east and 13 km north. Find the magnitude and direction of the resultant displacement vector.
(b) Each of the vectors A and B has a magnitude of 3 m. Calculate and show the
following vectors graphically:
i. A + B
ii. A − B
iii. B − A
iv. A − 2B
2. Vector operations: dot product
(a) Find the dot product w · v for the vectors w = î + 4ĵ − 2k̂ and v = −3î − 12ĵ + 6k̂.
i. What is the angle between the two vectors w and v?
ii. What is the angle between w and the z-axis?
(b) What is the component of the vector a = 3î + 4ĵ − 2k̂ along the vector b = î + ĵ?
In other words, what is the projection of a along b?
3. Vector operations: unit vectors, components and resolution of vectors
(a) Vectors A and B have components Ax = −5.0 cm, Ay = 1.1 cm, Az = −3.5 cm
and Bx = 8.8 cm, By = −6.3 cm, Bz = 9.2 cm along the x, y, z axes. Determine
the x, y, z components of the following vectors, and express these vectors in unit
vector notation:
i. A + B
ii. B − A
iii. 3B + 2A
(b) Two people pull on a stubborn mule as seen from the helicopter view in the
Figure. Find:
i. The single force which is equivalent of the two forces shown.
ii. The force exerted by the mule in order to not move from its position (that
is, to make the net force equal to zero).
4. Vector operations: cross product
The cross product between two three-dimensional vectors a = ax î + ay ĵ + az k̂ and
b = bx î + by ĵ + bz k̂ is determined by calculating the determinant:
î
ĵ k̂
a × b = ax ay az
bx by bz
Using the above procedure, calculate a × b and b × a for the vectors a = î − 2ĵ + k̂
and b = 2î − ĵ + 3k̂. What is the angle between a × b and b × a?
2
5. Vector operations: an alternative method to calculate cross product without using determinants
In this problem, you will learn an alternative way to calculate cross product between
two vectors using matrix multiplication.
The cross product between two three-dimensional vectors a = ax î + ay ĵ + az k̂ and
b = bx î + by ĵ + bz k̂ is determined by left-multiplying the vector b with a skewsymmetric matrix form of the vector a as shown below:

 
0
−az ay
bx
0
−ax  by 
a × b =  az
−ay ax
0
bz
⊤
The solution of the above procedure is obtained as a column vector cx cy cz ,
which can be simply expressed in the unit vector form as c = cx î + cy ĵ + cz k̂ can be
written as a column vector
Using the above procedure, calculate a × b and b × a for the vectors a = 3î − ĵ + 4k̂
and b = î + 4k̂. What is the angle between a × b and b × a?
6. Differentiation
(a) Chain rule: Find the derivative of g(t) = tan(5 − sin 2t).
dy
(b) Implicit differentiation: Find
if y 2 = x2 + sin (xy)
dx
7. Differentiation: velocity and acceleration
The position of an object moving in 3-D space at some time t is described by the
vector r = (t2 )î + (3t + 2)ĵ + (sin t)k̂. What are the velocity and acceleration vectors?
8. Differentiation: parametric function
Instead of a function y(x) being defined explicitly in terms of the independent variable
x, it is sometimes useful to define both x and y in terms of a third variable, t say,
known as a parameter. Such parametric functions are commonly seen in dynamics.
Suppose, you are given a parametric function y(x) in the form of y(t) and x(t). The
dy
derivative
can simply be found using chain rule as follows:
dx
dy
dy
= dt
dx
dx
dt
provided
dx
̸= 0
dt
The second derivative can also be found using another application of chain rule:
d dy
d2 y
d dy
dt dx
=
=
2
dy
dx
dx dx
dt
3
You are given some functions defined using some parameter t. For each of the following
dy
d2 y
functions, determine
:
and
dx
dx2
(a) x(t) = sin t , y(t) = cos t
(b) x(t) = 3t2 + 1 , y(t) = t3 − 2t2
(c) x(t) = e−t , y(t) = t3 + t + 1
(d) x(t) = 3t2 + 4t , y(t) = sin (2t)
9. Differentiation: radius of curvature
As objects move in the space, they may move along a curved path. The radius of
curvature ρxy of some path defined in x-y coordinates can simply be calculated using
the formula given below:
"
2 #3/2
dy
1+
dx
ρxy =
d2 y
dx2
The path of an object is defined by x(t) = 2t and y(t) = 3t2 − t. Calculate the path’s
radius of curvature ρxy .
10. Differentiation: maxima and minima of a function
dy
= 0 is defined as a stationary point. Now,
For a function y(x), any point where
dx
there are three possibilities for the double derivative:
d2 y
< 0 at the point.
dx2
d2 y
> 0 at the point.
• A stationary point must be a minima of the function if
dx2
d2 y
• If
= 0 at the stationary point, it may be a maximum or a minimum, or
dx2
neither. The second derivative test is inconclusive in this case.
• A stationary point must be a maxima of the function if
Using the procedure described above, determine the maxima and minima of the function y = 2x3 − 9x2 + 12x
11. Integration: graphical intuition: area under curve
A function f (x) = x2 − 4x + 3 is given. Evaluate the definite integrals:
R1
(a) 0 f (x) dx. Is this integral positive, negative or zero?
R3
(b) 1 f (x) dx. Is this integral positive, negative or zero?
R3
(c) 0 f (x) dx. Is this integral positive, negative or zero?
Now, sketch the graph f (x) vs. x of the function. Can you explain why the area
evaluated in the third case is zero?
4
12. Integration by substitution: substituting a differential form
Evaluate the following integrals:
R
(a) cos (3x + 4) dx
√
R sin x
√
(b)
dx
x
13. Integration by substitution: substituting a trigonometric function
Evaluate the following integral:
(Hint: substitute x = tan θ)
Z
1
dx
1 + x2
14. Integration by parts
Evaluate the following integral:
R
4x cos (2 − 3x) dx
5