Physics: An Introduction Chapter 2 -- Reference Tools & Resources

Chapter 2: One-Dimensional Kinematics
Reference Tools & Resources

Reference Tools & Resources

I. Key Terms and Phrases

mechanics: the study of how objects move and the forces that cause motion.

kinematics: the branch of physics that describes motion.

distance: the total length of travel.

displacement: the change in position of an object.

average speed: distance divided by elapsed time.

velocity: the rate of change of displacement with time.

tangent line: the straight line that intersects a curve at a point P as the result of a limiting process of secant lines through points surrounding P.

acceleration: the rate of change of velocity with time.

free fall: the motion of an object subject only to the influence of gravity.

the acceleration of gravity: the acceleration that results from Earth's gravitational pull.

II. Important Equations

Name/Topic Equation Explanation
displacement Dx = x_f - x_i Displacement is the change in position of an object.

average velocity v_av = Dx/Dt Average velocity is the displacement divided by the elapsed time.

constant velocity motion Dx = v Dt When velocity is constant, the displacement equals velocity times elapsed time. For this case v = v_av.

average acceleration a_av = Dv/Dt Average acceleration is the velocity change divided by the elapsed time

motion with constant acceleration
v = v₀ + at velocity changes linearly with time

position in terms of average velocity

position in terms of acceleration and time

velocity squared in terms of displacement

Name/Topic	Equation	Explanation
displacement	Dx = x_f - x_i	Displacement is the change in position of an object.
average velocity	v_av = Dx/Dt	Average velocity is the displacement divided by the elapsed time.
constant velocity motion	Dx = v Dt	When velocity is constant, the displacement equals velocity times elapsed time. For this case v = v_av.
average acceleration	a_av = Dv/Dt	Average acceleration is the velocity change divided by the elapsed time
motion with constant acceleration	v = v₀ + at	velocity changes linearly with time
		position in terms of average velocity
		position in terms of acceleration and time
		velocity squared in terms of displacement

III. Know Your Units

Quantity(ies) Dimension SI Unit
displacement [L] m

velocity, speed [L]/[T] m/s
acceleration [L]/[T²] m/s²

Quantity(ies)	Dimension	SI Unit
displacement	[L]	m
velocity, speed	[L]/[T]	m/s
acceleration	[L]/[T²]	m/s²

IV. Miscellaneous Tips

(A) Motion with Constant Acceleration

The four equations for constant acceleration form a set in which each equation has a key quantity missing. Taking them in the order listed in the table below, the first equation is missing displacement (x - x₀), the second is missing acceleration a, the third is missing final velocity v, and the fourth is missing time t. From this point-of-view, if an important quantity is unknown you have an equation that does not require it. However, you may have noticed that each equation contains the initial velocity v₀. Show that a fifth equation missing v₀ is

.

Use of this equation would have considerably simplified the solution to Exercise 2.8.

(B) Graphical Analysis

In the section on graphical analysis of accelerated motion it was pointed out that for constant acceleration the velocity-versus-time plot is linear and its slope equals the acceleration. Suppose, however, that you only have position-versus-time data. This plot is parabolic; is there any good way to get the acceleration from this set of data? The answer is yes if the motion starts from rest. For this special case of starting from rest the equation for x as a function of t reduces to

.

Notice that although x is quadratic in t, it is linear in t². So if you treat t² as a single variable and plot x-versus-t², you will get a linear curve. The slope of this line equals one-half of the acceleration. Just double the slope and you're done.