Describing an objects position and motion

using calculus concepts

Application of some higher order derivatives.

Analyzing the straight line motion of an object in front of a motion detector.

differentiatedifferentiate

Differentiate twice

Undo the derivative (integration, anti-derivative). Data point needed to determine the constant.

Undo the derivative (integration, anti-derivative). Data point needed to determine the constant.

¿𝑑𝑠(𝑡)𝑑𝑡

¿𝑑𝑣 (𝑡)𝑑𝑡

or  𝑑2𝑠 (𝑡)𝑑2 𝑡

   “s(t)” “v(t)” “a(t)”

Position at time “t” velocity at time “t” acceleration at time “t”changing position over time. Changing velocity over time.

Position relative to the motion detector

meters

Analyzing the motion of 5 objects in front of a motion detector

Time, “t” seconds

Object “A”

Object “B”

Object “C”

Object “D”

Obj

ect “

E”

Position is unchanged,Velocity is zero

s(t)= 5 m or v(t)=0 m/s

𝑠 (𝑡 )=0.5 𝑡+5 Straight line, constant velocity

m/s 𝑑𝑣 (𝑡)𝑑𝑡

𝑜𝑟 𝑎 (𝑡 )=0𝑚 /𝑠2then

𝑠 (𝑡 )=2.5 𝑡+5

𝑠 (𝑡 )=7 𝑡2+7

s (t )=−1 t+5

stee

per s

o m

ovin

g at a

faste

r con

stant

spee

d th

an o

bjec

ts A

, B an

d D

constant speed

𝑎 (𝑡 )=0𝑚/ 𝑠2

m/s 𝑎 (𝑡 )=0𝑚/ 𝑠2Constant slope, Constant velocity, no acceleration

Object “E” has positive changing tangent slopes, speed is changing m/s 𝑎 (𝑡 )=14𝑚/ 𝑠2

Constant velocityNo acceleration

𝑑𝑣 (𝑡 )𝑑𝑡

𝑜𝑟 𝑎 (𝑡 )=0𝑚/ 𝑠2

If the acceleration is positive, does that mean the object is speeding up? Not always.• Plot , 0≤t ≤7• Analyze the motion of the object. • Superimpose the velocity and acceleration graphs, • by first predicting their prediction• then by taking derivatives and accurately plotting their position

• Using the equations analyze the motion of the object at 4 seconds and then verify with the graphed data.• Determine the algebraic characteristics of an object that • is speeding up * Slowing down• Moving towards the detector * Moving away from the detector