Timedora

1 Kinematics

Learn how position, displacement, velocity, and acceleration describe motion, and how to apply kinematic equations, motion graphs, and component analysis.

Describe Motion with a Reference Frame

Kinematics describes motion without examining what causes it. A complete description begins by choosing a reference frame, a coordinate system, and a positive direction. Measurements of position and motion are always relative to that choice.

In one dimension, position can be represented by a coordinate such as xx. In two dimensions, position is a vector, r⃗=(x,y)\vec r=(x,y), with components along perpendicular axes. The origin is the reference point from which position is measured.

Position, , and Distance

Position tells where an object is relative to the origin. Its is the difference between its final and initial positions:

Δx=xf−xi\Delta x=x_f-x_i

For a vector position, the corresponding relation is Δr⃗=r⃗f−r⃗i\Delta\vec r=\vec r_f-\vec r_i. includes direction and depends only on the initial and final positions, not on the route taken.

Distance traveled is the total length of the path. It is never negative. For example, a person who walks 5 m5\ \text{m} east and then 2 m2\ \text{m} west travels a distance of 7 m7\ \text{m}, but has a of 3 m3\ \text{m} east.

Takeaway: Distance measures the whole path; compares only the endpoints.

and

Average is divided by elapsed time:

v⃗avg=Δr⃗Δt\vec v_{\text{avg}}=\frac{\Delta\vec r}{\Delta t}

Average speed instead uses distance traveled divided by elapsed time. It is nonnegative and may differ from the magnitude of average . Instantaneous describes motion at a particular instant; it is the rate at which position changes, and its magnitude is speed.

Average is the change in per unit time:

a⃗avg=Δv⃗Δt\vec a_{\text{avg}}=\frac{\Delta\vec v}{\Delta t}

Instantaneous is the rate of change of . It can change an object's speed, direction, or both. In one dimension, interpret the signs relative to the chosen positive direction: with the same sign as increases speed at that moment, while with the opposite sign decreases it.

For example, a cyclist moving initially at 4.0 m/s4.0\ \text{m/s} with 1.5 m/s21.5\ \text{m/s}^2 for 3.0 s3.0\ \text{s} reaches:

v=4.0+(1.5)(3.0)=8.5 m/sv=4.0+(1.5)(3.0)=8.5\ \text{m/s}

The during that interval is:

Δx=(4.0)(3.0)+12(1.5)(3.0)2=18.75 m≈19 m\Delta x=(4.0)(3.0)+\frac{1}{2}(1.5)(3.0)^2=18.75\ \text{m}\approx19\ \text{m}

Takeaway: describes how position changes; describes how changes.

Solve One-Dimensional Motion

When is constant throughout an interval, the following equations relate initial position x0x_0, final position xx, initial v0v_0, final vv, aa, and elapsed time tt:

v=v0+at,x=x0+v0t+12at2,v2=v02+2a(x−x0),x−x0=v0+v2 t.\begin{aligned} v &= v_0+at,\\ x &= x_0+v_0t+\frac{1}{2}at^2,\\ v^2 &= v_0^2+2a(x-x_0),\\ x-x_0 &= \frac{v_0+v}{2}\,t. \end{aligned}

Choose an equation that includes the known quantities and the unknown you need. Keep signs consistent with the chosen positive direction, and check that units match: position in metres (m\text{m}), in metres per second (m/s\text{m/s}), and in metres per second squared (m/s2\text{m/s}^2).

These should not be applied when changes over the interval.

Takeaway: Identify the knowns, the unknown, and whether is constant before selecting an equation.

Read

connect changes over time to physical quantities. On a position–time graph, the slope is . A straight line has constant , while a changing slope indicates changing . The slope of a tangent at a point gives instantaneous .

On a –time graph, the slope is , and the signed area between the graph and the time axis gives over the interval. A graph above the axis represents positive ; one below it represents negative .

On an –time graph, the signed area gives the change in . A horizontal line represents constant .

A negative slope or graph value does not automatically mean an object is slowing down. For example, negative and negative have the same sign, so speed increases at that moment.

Takeaway: Read slopes and areas in relation to the graph's axes, and interpret signs using the chosen positive direction.

Extend Kinematics to Two Dimensions

In two dimensions, position, , , and each have components along perpendicular axes:

r⃗=(x,y),v⃗=(vx,vy),a⃗=(ax,ay)\vec r=(x,y),\qquad \vec v=(v_x,v_y),\qquad \vec a=(a_x,a_y)

Analyze each component separately, using the same elapsed time for both. For constant , apply the one-dimensional equations independently to the horizontal and vertical components.

is the idealized motion of an object through the air when air resistance is neglected. Near Earth's surface, horizontal is zero, while vertical is approximately g=9.8 m/s2g=9.8\ \text{m/s}^2 downward. If upward is positive, then ay=−ga_y=-g. The horizontal and vertical motions share the same elapsed time, but their equations can be solved separately. The direction of may change during flight even when horizontal remains constant.

Takeaway: Separate motion into perpendicular components, keep a consistent sign convention, and use one shared time interval.