Motion Of A Particle
Motion
Motion of a particle moving along a straight line (one dimensional motion), ignoring
friction, gravity etc. The particle is any moving object – ball, car etc.
Displacement (x or s)
- Gives the position of a particle from a central position, the origin 0.
- x > 0, positive displacement – particle is to the right of 0.
- x < 0, negative displacement – particle is to the left of 0.
- x = 0, zero displacement – particle at 0.
- Displacement is a function of time. x  f  t  . Time t is always positive.
-
-
Initially t = 0.
Distance = displacement . Eg: displacement = - 2 ie: the particle is a distance
of 2m to the left of the origin. Distance travelled will always be positive or
zero.
Maximum displacement occurs when the velocity is zero (or at the endpoints
of the domain).

dx
)
dt
Is the instantaneous velocity. It measures the rate of change of displacement
with respect to time.
Gives the speed and direction of the particle.
v > 0, positive velocity – particle is moving to the right.
v < 0, negative velocity – particle is moving to the left.
v = 0, Zero velocity – particle is stationary, at rest, not moving.
Speed = velocity . Always positive.
Maximum velocity occurs when the acceleration is zero (or at the endpoints of
the domain).
Velocity (v or x or
-

d 2 x dv
v
or
or
)
dt
dt 2
The acceleration of an object is the rate at which the velocity is changing.
Positive acceleration – acceleration is acting to the right.
Negative acceleration – acceleration is acting to the left.
Zero acceleration – the particle’s velocity is not changing ie: it is constant.
Velocity and acceleration in the same direction – particle is speeding up.
Velocity and acceleration in opposite directions – particle is slowing down.
Acceleration (a or x or
-
Describing Motion
When describing motion give details about the following:
- Starting position and initial velocity
- When and where the object is at rest and the behaviour after rest
- Direction the object is travelling ie: velocity + or –
- Slowing down or speeding up
Examples:
1)
A particle is moving in straight line so that its displacement, x m, from a fixed
point 0 at time t secs, is given by x  t 2  11t  22 .
a)
Find the initial position of the particle.
b)
Find the displacement after 5 secs.
c)
At what times will the particle be 2m to the left of 0?
d)
Sketch x as a function of t.
2)
A particle is moving in a straight line. Its displacement is given by
x  8  12t  3t 2 . When and where is the particle at rest?
3)
A particle is moving with a velocity, x ms-1, given by x  14t  6t 2 . Find:
a)
the initial acceleration
b)
the initial displacement if the particle is 5m to the left of the origin
after 2 secs.
4)
A particle is moving with acceleration x ms-2 given by x  6t  6 (0  t  6) .
If the initial velocity is 5ms-1, find the greatest speed of the particle.
5)
An object with these measurements




when t = 0, s = -3, v =2,
when t = 1.5, s =0, v = 3.5
when t = 4, s = 2.5, v = 0
when t = 6.5, s = 0, v = -3.5
when t = 8, s = -4, v = -5
Describe this motion.
Solution to 5)
The particle starts 3 metres to the left of the origin and is moving towards the origin
(or in the positive direction) with a velocity of 2 m/s. After 1.5 seconds it passes the
origin with a velocity of 3.5 m/s still moving to the right. 4 seconds after the
beginning of timing the particle comes to rest 2.5 m on the positive side of the
origin, turns and increases speed as it moves to the left, towards the origin. At t =
6.5 seconds the particle passes the origin with a velocity of - 3.5m/s. 8 seconds after
the beginning of timing the particle is 4 metres to the left of the origin, moving
away from the origin with a velocity of -5 m/s.
v = 2 m/s
-4m
v = - 5 m/s
-3m
v = 3.5 m/s
O
v = - 3.5 m/s
v=0
2.5 m
Exercises:
1. A particle moves in a straight line so that its position x m from a fixed point O on
the line at any time t sec is given by x = t2 – 5t + 6 . Find:
a) its initial position
b) its initial velocity
c) when and with what velocity it first passes O,
d) when it passes through O a second time
e) when and where its velocity is zero.
2. The position s of a particle at time t is given by s = 4 – cos 2t.
a) Sketch the graph of s as a function of t in the domain 0  t  2,
b) Find the times when the particle is at rest,
c) the time when the particle first reaches the position s = 0.
3. A motorist takes 3 hours to cover the distance between two points A and B on a
country road. At time t hours after passing A, his speed in v km/h is given by
v = 60 + 40 e-t .
a) calculate his speeds when he passes points A and B.
b) sketch the velocity –time curve and comment on the particle’s motion for large
t.
4. A particle moves in a straight line and at time t minutes, its distance x cm, from a
fixed point O is given by x = 4 loge(1+t) – 2t, t  0
a) Find x when t = 0
b) Find the initial velocity
c) At what time is the velocity zero?
5. A body falls from rest in a medium and its velocity v m/s, at any time t s, is given
by
v = 49( 1-e-0.5t) . Find:
a)
b)
c)
d)
the acceleration at any time t,
the initial velocity,
the terminal velocity i.e. its velocity as t  ,
the distance fallen in the first 5 seconds.
6. The acceleration of a particle moving in a straight line is given by
d 2x
 12 cos 2t . Initially v =0 and x = 6 cm. Find its velocity and displacement at
dt 2
any time t seconds and sketch the graph of the displacement. How many times
does the particle change direction in the first 10 seconds?
7. A particle moves in a straight line so that at any time t  0 its velocity v is given
1
by v 
.
1 t
a) If its distance from a fixed point O in the line is x, show that x = loge(1+t),
given that x = 0 when t = 0.
b) Show that the particle is moving away from O as t increases.
c) Find the acceleration when t = 0.
8. A body is projected vertically upwards with an initial velocity of 30 m/s. It rises
with a deceleration of 10 m/s. Find:
a) its velocity at any time t,
b) its height, h m, above the point of projection at any time t;
c) the greatest height reached;
d) the time taken to return to the point of projection.