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1.3 Representing Motion

Syllabus
2024
Topic
1.3
Level
—

Learning objectives

1.3A—Describe the position, velocity, and acceleration of an object using representations of that object’s motionDescribe the position, velocity, and acceleration of an object using representations of that object’s motion.• Motion can be represented by motion diagrams, figures, graphs, equations, and narrative descriptions.• For constant acceleration, three kinematic equations can be used to describe instantaneous linear motion in one dimension: =+ vv atxx x 0 =+ + xx vt at1 2 2 xx 00 ()=+ − vv ax x2xx x 2 0 2 0 Note: The equations above are written to indicate motion in the x-direction, but these equations can be used in any single dimension as appropriate.• Near the surface of Earth, the vertical acceleration caused by the force of gravity is downward, constant, and has a measured value approximately equal to• Graphs of position, velocity, and acceleration as functions of time can be used to find the relationships between those quantities. TOPIC 1.3 Representing Motion- i. An object’s instantaneous velocity is the rate of change of the object’s position, which is equal to the slope of a line tangent to a point on a graph of the object’s position as a function of time.- ii. An object’s instantaneous acceleration is the rate of change of the object’s velocity, which is equal to the slope of a line tangent to a point on a graph of the object’s velocity as a function of time.- iii. The displacement of an object during a time interval is equal to the area under the curve of a graph of the object’s velocity as a function of time (i.e., the area bounded by the function and the horizontal axis for the appropriate interval).- iv. The change in velocity of an object during a time interval is equal to the area under the curve of a graph of the acceleration of the object as a function of time. BOUNDARY STATEMENT AP Physics 1 does not expect students to quantitatively analyze nonuniform acceleration. However, students will be expected to be able to qualitatively analyze, sketch appropriate graphs of, and discuss situations in which acceleration is nonuniform. BOUNDARY STATEMENT For all situations in which a numerical quantity is required for g, the value gm s10 / 2 will be used. However, students will not be penalized for correctly using the more precise commonly accepted values of == g or g 9.81 m/s9 .8 m/s.22 AP Physics 1: Algebra-Based Course and Exam Description Kinematics UNIT 1

Motion representations must tell the same story

One motion, several representations

A motion diagram, graph, equation, figure or narrative can describe the same motion. A valid translation preserves the signs, changes and time interval represented by position xx, velocity vxv_x and acceleration axa_x.

Choose an equation by what is known

Use when the unknown is… Constant-acceleration relationship
Final velocity after time tt vx=vx0+axtv_x=v_{x0}+a_xt
Position after time tt x=x0+vx0t+12axt2x=x_0+v_{x0}t+\tfrac12a_xt^2
Velocity after a displacement, with no time needed vx2=vx02+2ax(x−x0)v_x^2=v_{x0}^2+2a_x(x-x_0)

Read slopes and signed areas

Representation Read this feature It gives…
Position–time graph Tangent slope Instantaneous velocity
Velocity–time graph Tangent slope Instantaneous acceleration
Velocity–time graph Signed area over an interval Displacement
Acceleration–time graph Signed area over an interval Change in velocity

Check that the representations agree

Hypothetical example: a point object starts at x0=1 mx_0=1\,\mathrm{m} with vx0=0v_{x0}=0 and constant ax=+2 m s−2a_x=+2\,\mathrm{m\,s^{-2}} for t=3 st=3\,\mathrm{s}. Then vx=0+(2)(3)=+6 m s−1v_x=0+(2)(3)=+6\,\mathrm{m\,s^{-1}} and x=1+0+12(2)(32)=10 mx=1+0+\tfrac12(2)(3^2)=10\,\mathrm{m}. Its displacement is +9 m+9\,\mathrm{m}, equal to the area under its velocity–time graph.

Conditions and boundaries

The three equations require constant acceleration. AP Physics 1 treats nonuniform acceleration quantitatively only at a qualitative graph level. Near Earth, gravitational acceleration is downward with magnitude g≈10 m s−2g\approx10\,\mathrm{m\,s^{-2}}; its component is negative if upward is chosen positive.