Premed · Premed · Physics 1
Lecture 2: Kinematics in One Dimension
Physics I — Mechanics & Thermodynamics
Learning Objectives
By the end of this lecture, students will be able to:
- Define and distinguish between position, displacement, distance, speed, and velocity
- Calculate average and instantaneous velocity and acceleration
- Interpret and extract information from position-time and velocity-time graphs
- Apply the kinematic equations for constant acceleration to solve one-dimensional problems
- Analyze free-fall motion under gravity
Lecture Content
I. Position, Displacement, and Distance
Position (x) describes the location of an object along a coordinate axis relative to a chosen origin. Defining a position requires specifying a reference frame, including an origin and a positive direction. Displacement (Delta x) is the change in position, defined as Delta x = x_f - x_i. Displacement is a vector quantity, meaning it carries directional information and can be positive or negative along the axis. Distance, by contrast, is the total path length traveled and is always a scalar quantity that is non-negative. Distance is always greater than or equal to the magnitude of displacement. For example, if a person walks 3 m east and then 1 m west, the displacement is +2 m east, but the total distance traveled is 4 m.
II. Velocity
Average velocity is displacement divided by elapsed time: v_avg = Delta x / Delta t = (x_f - x_i) / (t_f - t_i). Because it is derived from displacement, average velocity carries directional information and can be positive or negative. Average speed, on the other hand, is distance divided by elapsed time. It is always non-negative and is not necessarily equal to the magnitude of the average velocity.
Instantaneous velocity is the velocity at a specific instant of time. Mathematically, it is defined as the limit v = lim(Delta t -> 0) Delta x / Delta t = dx/dt, which is the derivative of position with respect to time. Geometrically, the instantaneous velocity at a point on an x(t) graph corresponds to the slope of the tangent line at that point. Speed is simply the magnitude of instantaneous velocity: speed = |v|.
<image>Panel A: A position vs. time graph showing a curved path. A secant line between two points is labeled "average velocity = slope of secant" and a tangent line at one point is labeled "instantaneous velocity = slope of tangent." Panel B: The same motion shown on a number line with initial and final positions marked, displacement arrow, and total distance annotated along the path.</image>
III. Acceleration
Average acceleration is the change in velocity divided by elapsed time: a_avg = Delta v / Delta t = (v_f - v_i) / (t_f - t_i). Instantaneous acceleration is the rate of change of velocity at a specific instant: a = dv/dt = d^2x/dt^2. On a v(t) graph, instantaneous acceleration is the slope of the tangent line.
Acceleration is a vector quantity with both magnitude and direction. A common misconception is that positive acceleration always means speeding up, but this is not the case. What matters is the relationship between the signs of velocity and acceleration. If v and a have the same sign, the object speeds up. If they have opposite signs, the object slows down. The SI unit of acceleration is m/s^2.
IV. Kinematic Equations for Constant Acceleration
When acceleration is constant, four fundamental equations relate position, velocity, acceleration, and time: (1) v = v_0 + at, (2) x = x_0 + v_0 t + (1/2)at^2, (3) v^2 = v_0^2 + 2a(x - x_0), and (4) x = x_0 + (1/2)(v_0 + v)t. A reliable strategy for using these equations is to list all known quantities (x_0, x, v_0, v, a, t), identify the unknowns, and then select the equation that contains the unknown along with the knowns. It is best practice to solve algebraically before substituting numbers, and then to check that the units and sign of the answer are sensible. These equations are only valid for constant acceleration and should not be applied when acceleration varies with time.
<image>A summary card showing the four kinematic equations in a box. Beside each equation, a column lists which variable is absent from that equation (e.g., equation 3 does not contain t). Below, a flowchart: "What do you know?" leads to branches for different combinations of knowns, each pointing to the recommended equation to use.</image>
V. Graphical Analysis of Motion
Graphs of motion provide powerful tools for qualitative reasoning and for checking algebraic solutions. On a position vs. time graph x(t), the slope at any point gives the velocity. A straight line indicates constant velocity, while a parabolic shape (for constant acceleration) indicates uniformly accelerated motion. A curve that is concave up corresponds to positive acceleration, and one that is concave down corresponds to negative acceleration.
On a velocity vs. time graph v(t), the slope gives the acceleration, a straight line represents constant acceleration, and the area under the curve equals the displacement (Delta x = integral of v dt). On an acceleration vs. time graph a(t), the area under the curve equals the change in velocity (Delta v = integral of a dt), and for constant acceleration, a(t) appears as a horizontal line.
<image>Three vertically stacked graphs for the same motion (a ball thrown upward and caught on return). Top: x vs. t showing a symmetric parabola peaking at the highest point. Middle: v vs. t showing a straight line with negative slope crossing zero at the peak. Bottom: a vs. t showing a constant horizontal line at a = -9.8 m/s^2. Dashed vertical lines connect corresponding features across all three graphs, with annotations explaining the connections (e.g., "v = 0 corresponds to max x").</image>
VI. Free Fall
Free fall describes motion under the influence of gravity alone, with no air resistance. Near Earth's surface, the acceleration due to gravity is g = 9.80 m/s^2, directed downward. With the common convention of choosing upward as positive, the acceleration becomes a = -g = -9.80 m/s^2. The kinematic equations apply directly: v = v_0 - gt, y = y_0 + v_0 t - (1/2)gt^2, and v^2 = v_0^2 - 2g(y - y_0).
Several key features of free fall deserve emphasis. At the highest point of upward motion, the velocity is zero, but the acceleration is still -g. For an object launched and landing at the same height, the time to rise equals the time to fall, and the speed at landing equals the speed at launch. Crucially, g is independent of mass, meaning all objects fall at the same rate in the absence of air resistance. This was Galileo's great insight. Objects in free fall are not weightless; gravitational force still acts on them, providing the acceleration.
VII. Common Pitfalls and Problem-Solving Tips
Several common errors can derail kinematics problems. Velocity and speed are not the same thing, nor are displacement and distance. Always define a coordinate system and sign convention before attempting a solution. Remember that v = 0 does not imply a = 0, as illustrated by a ball at the top of its trajectory that momentarily has zero velocity but continues to accelerate downward at g. When an object changes direction, you can either split the problem at the turning point or use the full kinematic equations with consistent signs. Finally, quadratic equations in t may yield two solutions, and both should be interpreted physically. The non-physical solution, such as a negative time before the motion began, should be discarded.
