2. Solved Examples
1. Why does physics require extensive use of vectors as compared to just scalar quantities? Give a specific example where scalars are insufficient.
Solution:
Vectors are essential in physics because they describe quantities that have both magnitude and direction. Scalars only have magnitude. Many physical quantities, like displacement, velocity, acceleration, force, and momentum, require both magnitude and direction for a complete description.
Example: Consider two forces acting on an object. If you only know the magnitudes of the forces (scalars), you can't determine the net force without knowing their directions. If the forces act in the same direction, you add the magnitudes. If they act in opposite directions, you subtract. If they act at angles to each other, you need vector addition to find the resultant force. Scalars are insufficient in this case.
2. Given the two complex numbers z1 =2+3i and z2=5-3i, what is their sum? What is their difference.
Solution:
Sum:
Difference:
3. Given the complex numbers from problem 2, what is the angle between the x-axis and a line drawn from the origin to the point z1, if plotted in the complex plane?
Solution:
We need to find the argument of the complex number .
4. What is the magnitude (also called modulus) of each complex number in problem 2?
Solution:
Magnitude of z1:
Magnitude of z2:
5. What is the real part of each complex number in part 2? Imaginary part?
Solution:
z1 = 2 + 3i:
Real part: 2
Imaginary part: 3
z2 = 5 - 3i:
Real part: 5
Imaginary part: -3
6. Given two vectors and what is their sum? What is their difference?
Solution:
Sum:
Difference:
7. Given the same two vectors as in problem 6, what is the angle between the x-axis and
Solution:
8. What is the magnitude of each vector in problem 6?
Solution:
Magnitude of :
Magnitude of :
9. What is the x-component of each vector in problem 6? y-component?
Solution:
Vector :
x-component: 2
y-component: 3
Vector :
x-component: 5
y-component: -3
10. Given the vectors in problem 6, find and
Solution:
To find the unit vector, divide the vector by its magnitude.
11. Rewrite the vectors from problem 6 as and
Solution:
12. A velocity vector in 3D of a flying rock is given below. The m/s terms just stand for meters per second, which is the unit for velocity used in the SI system. Speed is the magnitude of the velocity. Find the speed of the rock and its direction of travel. Hint:
Solution:
First, find the speed (magnitude of the velocity):
Next, find the direction (unit vector):
13. Given the rock in problem 12, what is the angle between its velocity vector and each (x, y, and z) axis? Do this using direction cosines.
Solution:
The direction cosines are the components of the unit vector. If then , , and where is the angle with the x-axis, is the angle with the y-axis, and is the angle with the z-axis.
From the previous problem, . Thus
14. Given the vectors in problem 6, find if
Solution:
15. Find
Solution:
Without specifying a dot or cross product this is undefined.
16. Find
Solution:
This expression is undefined. You can't divide a scalar by a vector.
17. Impulse (J) is defined as the change in the momentum of an object, and is given by where is the force (measured in newtons) acting on an object and t is time (in seconds) over which the force acts. What is the change in the momentum of the object in the interval between t=0 and t=5 if the force is given by In the interval between t=5 and t=10?
Solution:
First interval (t = 0 to t = 5):
Second interval (t = 5 to t = 10):
18. When it comes to the study of ergonomics (study of interaction between humans and their environments), engineers need to, for instance, consider the comfort of passengers on trains. The sensation of being "jerked" around when throttle is applied suddenly is unpleasant. The amount of "jerking" can be quantified. It is literally called jerk in physics and is equal to the time derivative of the acceleration. Find the jerk associated with an acceleration vector of Then evaluate it at t=1.0. As mentioned, we will be careful to discuss units associated with all such terms soon.
Solution:
Jerk is the time derivative of acceleration:
Evaluate at t = 1.0:
19. What does it tell us about factors on which air drag depends if it is proportional to speed squared?
Solution:
If air drag is proportional to speed squared, it suggests that air drag depends on two separate factors that independently cause it to rise with speed. In this case it is the increased intensity of the collisions with the air molecules and also the increase in frequency of those collisions (think of passing through the gaps between air molecules more quickly).
20. If speed is tripled, how much larger will air drag become for an object? Show the math.
Solution:
Let the initial air drag be , where k is a constant that depends on the object's shape and air density. If the speed is tripled, the new speed is 3v, and the new drag force is .
Therefore, the air drag becomes 9 times larger.
21. What functional form do you expect to describe the motion of a vibrating membrane without damping and why?
Solution:
I would expect a sinusoidal (sine or cosine) function or a combination of such functions to describe the motion. This is because vibrating systems, without damping, exhibit simple harmonic motion. Simple harmonic motion is characterized by a restoring force that is proportional to the displacement, leading to oscillations described by sinusoidal functions.
22. Suppose a toy boat moves in a pool at at a speed given by v=1.0 meter per second at t=0, and that the boat is subject to viscous damping. The damping on the boat causes the rate of speed loss to be given by the expression How fast will the boat be traveling after 1 second? 3 seconds? 10 seconds? Use separation of variables to solve this.
Solution:
Separate variables:
Integrate both sides:
Solve for v:
, where
Use the initial condition v(0) = 1:
So, A = 1.
The equation for velocity is:
Now, evaluate at t = 1, 3, and 10 seconds:
t = 1 second:
t = 3 seconds:
t = 10 seconds:
23. Find (2.0x105)(3.6x104)/5.3x106 without using parentheses in your calculator. You need to know exactly how it needs to be typed into a calculator or into a program like MATLAB.
Solution:
In MATLAB:
2.0e5*3.6e4/5.3e6
On a scientific calculator (assuming one with an "EE" or "EXP" button):
2.0EE5 * 3.6EE4 / 5.3EE6
The key is to use the "EE" or "EXP" button, which tells the calculator to treat the following number as the exponent of 10. This avoids ambiguity in the order of operations. Nearly every programming language uses either 'E' or 'e' (or will allow either one).
24. Why can't this expression be true:
Solution:
This expression is not valid because it is an incorrect expression for something called torque (using the Greek lower case tau) that we'll get to later in the course and which is defined as the cross product of the position vector and the force vector , which is written as . The given expression implies an undefined form of multiplication of two vectors. We know this cannot be a correct expression simply because conventional multiplication of vectors is undefined. The cross product, by the way, results in a torque vector that is perpendicular to both and , representing the rotational effect of a torque.
25. Why is this not a correct equation:
Solution:
The correct equation is . The given equation is missing the vector symbol over the acceleration. Force and acceleration are vector quantities (they have both magnitude and direction), while mass is a scalar. Therefore, the acceleration must be a vector for the equation to be dimensionally and physically correct.
26. What's wrong with this expression:
Solution:
Kinetic energy is a scalar quantity, and it's calculated using the square of the magnitude of the velocity vector, not the square of the vector itself. The correct expression is or , where v is the speed (magnitude of velocity). Squaring a vector as written in the original expression is undefined.
27. Why can't this be correct:
Solution:
The expression is incomplete because it only specifies the magnitude of the acceleration (7 m/s2) but not its direction. Acceleration is a vector quantity, so it must have both magnitude and direction. A correct representation would include a direction, for example, or similar.