SPI Physics · Kinematics and Motion

Motion Graphs and Equations of Motion

The shape of a motion graph tells you the kind of motion, the slope and area of a velocity-time graph give acceleration and distance, and three equations link speed, time and distance under steady acceleration.

All Kinematics and Motion notes

Why this matters

Four questions in the bank, two of them from the 2022 paper: which graphs show constant speed, what the area under a velocity-time graph gives, a two-car velocity-time graph (the only HARD one), and how fast a monkey must jump to reach a branch. Two are EASY, one MODERATE and one HARD.

Concept 1 of 3: What the shape of each motion graph means

Before reading any numbers, look at two things: what is on the vertical axis, and whether the line is flat, sloping or curved. A flat line on one graph means something very different from a flat line on another.

Definition

On a distance-time graph, a straight sloping line means constant speed; a flat line means the body is at rest; a curve bending upwards means it is speeding up.
On a speed-time or velocity-time graph, a flat line means constant speed (or velocity); a straight sloping line means constant acceleration, so the speed is changing.
On an acceleration-time graph, a flat line above zero means steady acceleration, so the speed keeps rising.
Read the axis label first. The same flat line means rest on a distance-time graph and steady motion on a velocity-time graph.

Graph	Line shape	What the motion is
Distance against time	Flat	At rest
Distance against time	Straight, sloping up	Constant speed
Distance against time	Curving upwards	Speeding up
Velocity against time	Flat (above zero)	Constant velocity, so constant speed
Velocity against time	Straight, sloping up	Constant acceleration, speed changing
Speed against time	Straight, sloping up	Speed rising steadily
Acceleration against time	Flat (above zero)	Constant acceleration, speed changing

Constant speed shows up as a sloping distance-time line or a flat speed or velocity-time line.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q81Easy

Example 1 · Kinematics and Motion · Motion Graphs and Equations of Motion

Which of the following four graphs represent(s) motion at constant speed?

Reading a flat acceleration line as constant speed

A flat line on an acceleration-time graph means the acceleration is steady, so the speed is changing all the time. Constant speed needs zero acceleration.

Reading a flat distance-time line as constant speed

A flat distance-time line means the body is at rest. Constant speed on that graph is a straight line that slopes.

Concept 2 of 3: Slope and area of a velocity-time graph

A velocity-time graph holds two answers. How steeply it climbs tells you how fast the velocity changes, which is acceleration. The space under it is velocity times time, which is how far the body went.

Definition

Slope of a velocity-time graph $=$ change in velocity $\div$ time taken $=$ acceleration.
Area under a velocity-time graph $=$ distance travelled (the displacement, for motion in one direction).
Split the area into rectangles and triangles: a rectangle is base $\times$ height, a triangle is $\frac{1}{2} \times$ base $\times$ height.
A line that is rising but flattening means the velocity is still increasing, at a decreasing rate; a falling line means slowing down.

Velocity-time graph

a = \text{slope} = \frac{v - u}{t} \qquad \text{distance} = \text{area under the graph}

Worked example

A car starts from rest and speeds up steadily to 24 m/s in 8 s, then keeps 24 m/s for 5 s more. From its velocity-time graph, find its acceleration in the first 8 s and the total distance.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 2 · Kinematics and Motion · Motion Graphs and Equations of Motion

The diagram shows the velocity - time graph of two moving cars P and Q. The graph indicates that

Mixing up slope and area

The slope of a velocity-time graph is the acceleration; the area under it is the distance. Area is never the acceleration.

Comparing two bodies by their speed at one moment

Which of two cars is ahead depends on the distance each has covered, which is the area under each graph up to that time, not on which one is faster at that instant.

Drill 1 more on slope and area of a velocity-time graph

Concept 3 of 3: The three equations of motion and a body thrown upwards

When acceleration stays the same, five quantities are tied together: starting speed u, final speed v, acceleration a, time t and distance s. Each equation leaves one of them out, so pick the one without the quantity you were not given.

Definition

The three equations hold only for constant acceleration.
A body thrown straight up slows down at g (about $9.8\ \text{m/s}^2$ , often taken as 10). At the highest point its speed is zero.
So to just reach a height h, the starting speed u satisfies $0 = u^2 - 2gh$ , that is $u^2 = 2gh$ .
A body dropped from rest has u $= 0$ and speeds up at g.

Equations of motion

v = u + at \qquad s = ut + \tfrac{1}{2}at^2 \qquad v^2 = u^2 + 2as \qquad u = \sqrt{2gh} \text{ to reach height } h

ustarting velocity
vfinal velocity
aacceleration
sdistance (displacement)
ttime

Worked example

A ball is thrown straight up at 20 m/s. Find the greatest height it reaches and the time it takes to get there. Take

g = 10\ \text{m/s}^2

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q87Moderate

Example 3 · Kinematics and Motion · Motion Graphs and Equations of Motion

A monkey was seen jumping up from ground level on to a branch 2.5m high. The initial velocity of the monkey is____ m/s.

Forgetting the speed at the top is zero

A body thrown up stops for an instant at its highest point. Put v = 0 there; that is what turns

v^2 = u^2 - 2gh

into

u = \sqrt{2gh}

Using the equations when acceleration changes

The three equations need steady acceleration. On a curved velocity-time graph, use the area and slope instead.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Slope and area of a velocity-time graph
Velocity-time graph
$a = \text{slope} = \frac{v - u}{t} \qquad \text{distance} = \text{area under the graph}$
The three equations of motion and a body thrown upwards
Equations of motion
$v = u + at \qquad s = ut + \tfrac{1}{2}at^2 \qquad v^2 = u^2 + 2as \qquad u = \sqrt{2gh} \text{ to reach height } h$

Reference tables (1)

What the shape of each motion graph means7 rows

Graph	Line shape	What the motion is
Distance against time	Flat	At rest
Distance against time	Straight, sloping up	Constant speed
Distance against time	Curving upwards	Speeding up
Velocity against time	Flat (above zero)	Constant velocity, so constant speed
Velocity against time	Straight, sloping up	Constant acceleration, speed changing
Speed against time	Straight, sloping up	Speed rising steadily
Acceleration against time	Flat (above zero)	Constant acceleration, speed changing

Constant speed shows up as a sloping distance-time line or a flat speed or velocity-time line.

Watch out for (6)

Reading a flat acceleration line as constant speed
→ What the shape of each motion graph means
Reading a flat distance-time line as constant speed
→ What the shape of each motion graph means
Mixing up slope and area
→ Slope and area of a velocity-time graph
Comparing two bodies by their speed at one moment
→ Slope and area of a velocity-time graph
Forgetting the speed at the top is zero
→ The three equations of motion and a body thrown upwards
Using the equations when acceleration changes
→ The three equations of motion and a body thrown upwards

Test yourself on a real paper

Sit a past SPI paper, timed and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Sit an SPI past paper Drill all 4 questions