Mechanics
Velocity-Time Graphs
Objective: Be able to represent and
interpret an object’s motion under
uniform acceleration graphically.
M1
We can draw a graph of an object’s velocity against time:
Acceleration is the rate of change of
velocity, therefore the gradient of
the line.
v
velocity
(ms-1)
The area under the line is a
trapezium, given by:
u
area =
displacement
u  v 
2
t0
time (s)
t1
t
Therefore the area gives the
displacement, s.
Eg. Draw a line on the axes below representing the motion of a particle accelerating
from 10 ms-1 to 60 ms-1 over 15 seconds.
What is the acceleration of this particle?
How far does this particle travel over the 15 seconds described?
velocity
(ms-1)
time (s)
1. Draw a V-t graph to represent a particle accelerating from 8ms-1 to 14ms-1 over 3
seconds.
2. Draw a V-t graph to represent a particle accelerating at 1.5ms-2 to a velocity of
18ms-1 after 6 seconds.
3. Draw a V-t graph to represent a particle accelerating at 2ms-2 from 4ms-1 to 12ms-1.
velocity
(ms-1)
velocity
(ms-1)
time (s)
velocity
(ms-1)
time (s)
time (s)
Eg.
The velocity-time graph shown is for a body which starts from rest,
accelerates uniformly to a velocity of 8ms-1 in 2 seconds, maintains that
velocity for 5 seconds, and then decelerates uniformly to rest. The entire
journey takes 11 seconds. Find:
a) the acceleration of the body during the initial part of the motion,
b) the retardation of the body during the final part of the motion,
c) the total distance travelled by the body.
8
velocity
(ms-1)
0
2
time (s)
7
11
Eg.
A particle is initially at rest at a point A on a straight line ABCD. The particle
moves from A to B with uniform acceleration, reaching B with a speed of
12 ms-1 in 2 seconds. The acceleration then alters to a constant 1 ms-2 and, 8
seconds after leaving B, reaches C. The particle then retards uniformly to
come to rest at D after a further 10 seconds.
a)
b)
c)
d)
e)
Draw a velocity-time graph for the motion and, from it, find:
the acceleration of the particle from A to B,
the speed of the particle at C,
the retardation of the particle from C to D,
the total distance travelled from A to D.
Eg.
A train travels along a straight piece of track between two stations, A and B.
The train starts from rest at A and accelerates uniformly at 1.25 ms-2 until it
reaches a speed of 20 ms-1. It then travels at this steady speed for a distance
of 1.56 km before decelerating at 2 ms-2 to come to rest a B. Find:
a) the distance from A to B,
b) the total time for the journey,
c) the average speed for the journey.
Eg.
A-Level Question
A train, travelling along a straight, horizontal track, has a steady speed of
18 ms-1 as it passes the point A. 15 seconds later it begins to slow down at a
uniform rate for 30 seconds until its speed is 10 ms-1. The train then
increases its speed for 45 seconds until it reaches a speed of 20 ms-1 as it
passes the point B.
a) Draw a sketch of the V-t graph for the journey of the train from A to B.
b) Calculate the acceleration of the train just before it reaches B.
c) Find the distance from A to B.