Finding Displacement Using V vs t Graphs - Kinematics

Displacement Plots — finding displacement from V vs t graphs

Warm-Up

This picture is of a single car moving across a room. The picture shows its position at equal time intervals. Describe its velocity.

Car moving across a room at equal time intervals

Video Instruction

showmethephysics.com

V. Finding Displacement Using V vs t Graphs

Velocity vs time plot, area under the curve highlighted

Since V = Δd/t

then Δd = Vt

= Area under the plot

= displacement (distance)

What is the displacement this object covered from t = 0 to t = 6 seconds?

↓

d = Area of the triangle

= ½bh

= ½(6 sec)(70 m/sec)

= 210 m

Ex 1

Object's displacement from t = 1 sec to t = 3 sec?

↓

Velocity vs time plot, rectangle region shaded

d = Area
= base × height

= (2 sec)(10 m/sec)

= 20 m

Ex 2

Object's displacement from t = 6 sec to t = 8 sec?

↓

Velocity vs time plot, trapezoid region shaded

d = area of trapezoid

= ½(h₁ + h₂) × base

= ½(25 m/s + 10 m/s)(2 sec)

= 35 m

Ranking Task

Smallest displacement?

Flash showmethephysics.com

Ranking Task Answer — Largest B, Smallest D

AP Question

Ex 3

a) Find displacement after 4 seconds
b) At what time does the object return to its starting point?

Velocity vs time plot, positive and negative regions

↓

After 4 seconds

½(3 sec)(−10 m/s) + ½(1 sec)(10 m/s)

−15 m + 5 m

= −10 m

Answer: 5 seconds when d = 0

Ex 4

When did car A catch up to car B?

Velocity vs time plot for car A and car B

↓

Car A caught up to car B when their d are equal

At 60 sec:

Car B: d = 60 m/s(30 sec) = 180 m

Car A: d = ½(60 m/s)(60 sec) = 180 m

Check Your Understanding

5 quick questions — pick an answer to see if you’re right.

1. On a velocity vs. time graph, what does the area under the line represent?

2. An object moves at a constant 10 m/s from t = 1 s to t = 3 s. What is its displacement?

3. An object speeds up uniformly from 0 to 30 m/s in 4 s. What is its displacement?

4. An object's velocity changes uniformly from 10 m/s to 30 m/s over 2 s. What is its displacement?

5. On a V vs. t graph, the area above the time axis is 5 m and the area below is 15 m. What is the net displacement?

NYSSLS Regents Practice · Kinematics · Mathematical Models

A Train Between Two Stations

A commuter train travels on a straight track between two stations. Graph 1 shows the train’s velocity from the moment it leaves the first station until it stops at the second station.

Graph 1: Train Velocity vs. Time

What is the distance between the two stations?

▶ Want a challenge? Open 6 more Regents-style questions (harder · mixed formats)

Challenge score: 0 / 6

Multiple choice

C1. Which list ranks the train’s displacement during each part of the trip from greatest to least?

Substitute values

C2. Substitute numerical values from Graph 1 into a mathematical representation that can be used to determine the distance the train travels while slowing down (80 s to 120 s). Then calculate the distance.

d = 12( s)( m/s) = m

Support or Refute · Justify

C3. A student makes the following claim.

Claim: The magnitude of the train’s acceleration while speeding up was twice the magnitude of its acceleration while slowing down.

Place a check mark in either the Support or Refute box. Justify your response using a mathematical relationship.

Support Refute

Determine the value

C4. Determine how far the train is from the first station at t = 100 s.

d = m

Multiple choice

C5. Planners want the same 1800-m trip to take only 100. s. The train will still speed up for 20. s and slow down for 40. s. What top speed is needed?

Constructed response

C6. Engineers propose installing stronger brakes so the train slows from 20 m/s to rest in 20 s instead of 40 s, still covering 1800 m. Using Graph 1, determine how much time this would save on the trip, and identify one drawback of this design change.

Free Fall →

Bonus

Amusing Surface Tension Experiment