Displacement and Constant Velocity: Different Approaches

 Physics Index

Where are we going with this? The information on this page connects to standards such as:

• Investigate and evaluate the graphical and mathematical relationship (using either manual graphing or computers) of one-dimensional kinematic parameters (distance, displacement, speed, velocity, acceleration) with respect to an object's position, direction of motion, and time.

•  Algebraically solve problems involving constant velocity and constant acceleration in one- dimension.

Displacement and Constant Velocity: Different Approaches
(Hmmm… Do we need more than one approach?

So, working with motion, we end up with a couple of similar formulas.

Δd = v • Δt

This means displacement equals average velocity times time.

df = di + Δd

This means final position equals initial position plus displacement.

Combining them gives:

df = di + v • Δt

Direction matters. You choose what is positive (for example, east or right). Units must be consistent (for example, meters and seconds).

What the formulas mean (plain-English + variables)

• $d_i$: initial position (displacement) relative to some reference point (the origin you choose).

• $d_f$: final position relative to the same reference point.

• $\Delta d$: displacement during the interval (net change in position). Positive/negative shows direction.

• $\mathrm{<span\; style="font-weight:\; bold;"><i>v</i>\; : <strong\; data-end="513"\; data-start="493"\; style="display:\; inline\; !important;">average\; velocity </strong>over\; the\; interval\; (displacement\; per\; unit\; time;\; includes\; direction).<br> <br>NOTE:\; Sometimes\; average\; velocity\; is\; show\; as\; <i>v<span\; style="font-size:\; xx-small;">(ave)</span></i></span>}$

• $\Delta t$: elapsed time for the motion.

Faced with word problems, it is convenient to know which approach to take. Which formula arrives at the answer most directly! (Cause, like who want's to do extra work!)

How to know which formula to use

If you are given or want displacement without mention of any points of reference, use:

Δd = v • Δt.

If the word problem includes mention to some reference point, then you need to use this:


df = di + v∆t

Example problems and solutions

Example A – Using Δd = v • Δt

Problem: A cyclist rides west at an average velocity of –4.0 m/s for 90 s. What is the displacement?

Why this formula? We need displacement, and we are given velocity and time.

Solution: 

Δd = v • Δt
∆d= (–4.0 m/s)(90 s)
∆d= –360 m.

Answer: The displacement is –360 m (360 m west).

Example B – Using df = di + v • Δt

Problem: A boat starts 50 m west of a dock (so di = –50 m if east is positive). It cruises with an average velocity of +2.5 m/s for 3.0 minutes. Where is it at the end?

Why this formula? We need final position and we know initial position, velocity, and time.

Solution:

Convert 3.0 minutes to 180 s. 

df = di + v • Δt
df = –50 + (2.5)(180)
df = –50 + 450
df = +400 m.

Answer: 400 m east of the dock.