Frequently Made Errors in Mechanics: Kinematics

Kinematics is the subset of dynamics that only concerns itself with time, displacement, velocity, and acceleration. A problem is a kinematics problem if it involves those quantities, but consideration of masses, forces, and work are unnecessary.

1. Scalars and Vectors

X  “A scalar only has magnitude while a vector has magnitude and direction”

 X “A scalar is a one-dimensional physical quantity, i.e. one that can be described by a single real number (sometimes signed, …”

✓ “A scalar is a one-dimensional quantity, i.e. one that can be described by a single real number (maybe signed) or a complex number…”

A magnitude is unsigned.  For a real number, the magnitude is the absolute value.

In mathematics, all real numbers and complex numbers are scalars, so a scalar can be negative, or might need two real numbers to express it.

In physics, some scalar quantities can definitely be signed (time, charge, potential).  Others (mass) cannot so far as we know, but even then it is not intrinsic in the definition.

2. Distance, Area*, Volume, and Speed

Most authorities define these as magnitudes:

“The scalar absolute value (magnitude) of velocity is called speed”

Arguably, this is unfortunate.  It makes it meaningless to discuss relative speed, etc.  It also makes this question ill-defined:

“A car travels on a straight road at speed u for time t and at speed u’ for time t’.  How far is it from its starting point?”

According to the ‘speed is a magnitude’ view, we cannot tell whether the car was reversing for one stage.

Some use the term pseudoscalar for a one-dimensional signed quantity.

For some purposes, it may be more convenient to define your variables as signed scalars, or, equivalently, as one-dimensional vectors.

For magnitudes, the operations of addition and subtraction are not associative.  If I’m traveling at 10kph I can speed up by 5kph then slow down by 12kph, but not in the other order.

*Note: An exception is the area element vector in a surface integral.  Something similar may apply to volumes in higher dimensions.

3. Distance versus Displacement

“A runner takes time t to run halfway around a circular track radius R.  What is the magnitude of the runner’s average velocity?”

Attempted answer:

X Distance = ##\pi R##, answer = ##\frac{\pi R}{t}##

That would have been correct had the question asked for the average speed.

Velocity is a vector, and the average velocity is given by ##\hat {\vec v}=\frac{\Delta \vec r}{\Delta t}##, where ##\Delta \vec r## is the displacement vector.  In the present case, the runner ran a straight-line distance 2R, so that is the magnitude of the displacement vector:

✓ Displacement = 2R, Average velocity =##|\hat {\vec v}|=\frac{|\Delta \vec r|}{|\Delta t|} = \frac{2R}t##

4. Average versus Instantaneous

 “A ball takes time t to roll down a ramp length D.  How fast is it moving at the bottom?”

Attempted answer:

X Speed = distance over time = ##\frac Dt##

Total distance over total time gives the average speed.  The instantaneous speed at a given point may be different.

Assuming constant acceleration down the ramp, the final speed can be obtained from the SUVAT equations.

5. Average of given speeds

“A cyclist cycles up a hill at speed u, then back down at speed 3u.  What is the cyclist’s average speed for the round trip?”

Attempted answer:

X ##\frac{u+3u}2 = 2u##

This is a form of the well-known solecism of taking an average of an average.

Suppose the length of the hill is D.  The time to climb the hill is ##\frac Du##, and the return time is ##\frac D{3u}##, for a total of ##\frac Du\left(1+\frac 13\right)= \frac {4D}{3u}##.  The average speed is total distance / total time =

✓ ##\frac{2D}{\frac {4D}{3u}} = \frac 32 u##.

6.Velocity versus Acceleration

X “If a particle is stationary it is not accelerating”

It depends on what is meant by stationary here.  If stationary for an extended time then clearly it is not accelerating, but if it is only stationary instantaneously (i.e. zero velocity at that instant) then it is accelerating.  A stone thrown straight up has zero velocity at its highest point, but its acceleration is g throughout.

7. SUVAT

The “SUVAT” equations only apply when the acceleration is known to be constant.  The acronym refers to the five standard variable names the equations use:

If any three of these are known, the remaining two can be determined.  Thus, there are five SUVAT equations, each omitting one variable:

To apply SUVAT equations to a problem:

1. Check that acceleration can be assumed constant.
2. Identify the variable to be determined.
3. Identify three other variables whose values are known, or which will feature in other equations related to the problem.
4. Select the equation involving those four variables.

Example: “A ball is thrown from ground level at angle ##\theta## above the horizontal at speed v.  How far away does it land?”

1. The acceleration is constant in both the vertical direction (g) and the horizontal direction (0).
2. The horizontal displacement is to be determined.
3. 1. In the horizontal direction, only the acceleration and initial velocity are known.
   2. In the vertical direction, the acceleration, initial velocity, and displacement are known.
   3. The time taken is common to the two directions.
4. 1. Find t from u, a, s in the vertical direction
   2. Find s from u, a, t in the horizontal direction.

X “The final vertical velocity is zero because it hits the ground”

Its vertical velocity will be zero after hitting the ground, but the process of hitting the ground involves a large upward acceleration.  Applying a SUVAT equation across the period from before to after impact is therefore not valid.  The ‘final’ velocity here means at the instant before impact.

8. Relationship between Kinematics Equations and Dynamics equations

Two of the equations in kinematics are just dynamics equations with mass cancelled out.