MOTION ALONG A STRAIGHT LINE - Penditamuda's Blog · 2011-01-30 · Additional Mathematics Motion along a straight line [email protected] 3 B. DETERMINE THE TOTAL DISTANCE TRAVELLED

Additional Mathematics Motion along a straight line

NR/GC/ Addmaths SMSJ/2009

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MOTION ALONG A STRAIGHT LINE

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MOTION ALONG A STRAIGHT LINE

1. DISPLACEMENT

A. IDENTIFY DIRECTION OF DISPLACEMENT OF A PARTICLE FROM

A FIXED POINT

NOTES:

If the right side of O is considered the positive direction, then

EXERCISE 1 A particle moves along a straight line with the displacement s m and t is the time after

passing through a fixed point O. Find the displacement of the particle after the corresponding

time.

Displacement formulae

Displacement within 1 s

Displacement at 3 s

s = t² -2t

s = 1² -2(1)

=-1

Meaning that the particle is 1m

on the left of O

s = 3² -2(3)

= 3

Meaning that the particle is

3m on the right O

(a) a) s = t² + 2

(b) s = t² -t -1

(d) s = t³ - 2t² -3

DISPLACEMENT

ORIENTATION

1.POSITIVE

The particle is on the RIGHT of O.

2.NEGATIVE

The particle is on the LEFT of O

3.ZERO

The particle is AT O or return to O again


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B. DETERMINE THE TOTAL DISTANCE TRAVELLED BY A PARTICLE

OVER A TIME INTERVAL

Example

A particle moves along the straight line with the displacement s m and t is the time

after passing through the fixed point O. Given the displacement s = 8 4t² therefore

find the total distance taken after 4 seconds.

Solution

When, t = 0, s = 8 4t²

s = 8 4(0)2

= 8

When t = 4, s = 8 4(4)²

= 8 64

= 56

Total distance traveled = 8 + 56 = 64m

O t = 0 t = 4

8 m 56 m


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EXERCISE 2

1. A particle moves along the straight line with the displacement s m

and t is the time after passing through the fixed point O. Find the total distance taken

after 3 seconds for the following cases.

Displacement formulae

initial displacement

(t = 0)

Total distance taken in the first three second

(a) s = 2t² 3

(b) s = 5 – 2t²

(c) s = 5t 1

(d) s = t2 – 5t


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EXERCISE 3

1. A particle moves along the straight line with the displacement s m is given as

s = t2 –3 and t is the time after passing through the fixed point O.

Find

a) the displacement of the particle at t = 1 and t = 3,

b) the distance traveled in the first 3 seconds


s = t2 – 4t + 3 and t is the time after passing through the fixed point O.

Find

a) the initial displacement ,

b) the total distance traveled in the third second,

c) the range of time when the particle is at the left of O .


s = 60 t – 5t2 and t is the time after passing through the fixed point O.

Find

a) the time when the particle is at 100 m to the right of O,

b) the time when the particle is at 225 m to the left of O,

c) the time when the particle passes through O again.

4. A particle moves along the straight line. Its displacement , s m from a fixed point O at

t second is given by s = 8t2 + t.

Find the total distance traveled

(a) in the first 5 seconds


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(b) in the fifth second.

5. A particle moves along the straight line. Its displacement , s m from a fixed point O at

t second is given by s = 5 + 4t – t2 . Given that the particle moves to the right of O until

t = 2 seconds and then moves to the left towards O.

Find the total distance traveled by the particle in the first 8 seconds.


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2. VELOCITY

A. DETERMINE VELOCITY FUNCTION OF A PARTICLE BY

DIFFERENTIATION

NOTES

The velocity of a particle, v, at the instant t is the rate of change of displacement

with respect to time, that is

v = dt

ds

If the direction of motion to the right is considered as the positive direction, then

Velocity (v) Particle moving to

Positive , v > 0

The particle is moving to the RIGHT

Negative, v < 0

The particle is moving to the LEFT

v = 0

The particle is at instantaneous rest/ stops

momentarily/ stationary/

maximum or minimum displacement

EXERCISE 4

A particle moves along a straight line with its displacements, s m and the time t after

passing through point O. Find the velocity when t = 3 and initial velocity for each of

the following:

Displacement velocity Velocity when t = 3 Initial Velocity

(a) s = 4 + 9t – 3t2 v = 9 – 6t v = 9 – 6(3) = –9 t = 0 , v = 4

(b) s= t² 4t + 2

(c) s = t² 4t

(c) s = t² -12t + 16


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Example

A particle moves along the straight line. Its displacement , s m from a fixed point O at

t second is given by s = = t² -12t + 16.

Find

(a) the time when the particle is at instantaneous rest

(b) the range of t for the positive velocity

Solution

s = t² -12t + 16

v = dt

ds= 2t – 12

(a) If particle is at rest ,v = 0,

2t –12 = 0

t = 6s

(b) If positive velocity, then v > 0;

2t –12 > 0

t > 6s

EXERCISE 5

A particle moves along a straight line with the velocity of v 1ms and t is the time after

passing a fixed point O. Find the time when the particle is at instantaneous rest and the time

as the particle moves to the left.

Velocity formulae Time when the particle comes

instantaneously to rest

the range of t when particle

moves to the left

(a) v = 9t² – 4

(b) v = t² –3t

(c) v = 2t2 + t – 28


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EXERCISE 6

1. A particle moves in a straight line with its displacement, s meter and time t second after

passed through a fixed point O. Given that s=2t³ – 15t² + 36t, find

(a) the displacement when the velocity is zero

(b) time when the particle moved to the left.

2. A particle moves in a straight line with its displacement, s meter and time t second after

passed through a fix point O. Given that s = 4 + 9t – 3t2, find

(a) the time when the particle comes instantaneously to rest

(b) the maximum displacement.

B. DETERMINE DISPLACEMENT FUNCTION OF A PARTICLE WHEN

VELOCITY IS GIVEN BY INTEGRATION

EXAMPLE:

The velocity of a particle which is moving along a straight line is given as v =3t + 4.

Find the displacement at 2 second.

SOLUTION:

s = vdt , s = dtt )43(

= ,42

3 2

ctt

c is a constant

When t = 0, s = 0;

0 = c )0(42

)0(3 2

c = 0

Therefore s = tt

42

3 2

when t =2 , s = )2(42

)2(3 2

= 14 m


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EXERCISE 7

VELOCITY FUNCTION Displacement when t = 3

1. v = 5 + 3t2

2. v = 8t3

4

EXERCISE 7:

A particle moves along a straight line with the velocity of v 1ms , and t is the time after

passing a fixed point O. Find the total distance traveled in the first three seconds:

Velocity displacement Time when the particle

stops momentarily

Total distance traveled in the

first three seconds

v = 6 – 6t s = dt)t66(

= 6t – 3t2 + c

t = 0, s = 0 , c = 0

s = 6t – 3t2

v = 0

6 – 6t = 0

t = 1

t = 0 , s = 0

t = 1 , s = 6(1) – 3(1)2 = 3

t = 3, s = 6(3) – 3(3)2 = –19

total distance = 3 + 19 = 22

v = 3t2 – 5t – 2

v = 2t – t2



EXERCISE 8

1. A particle moves in a straight line with the velocity v = 36 – 6t where t is the time in

second after passing through point O. Find

(a) the time when the distance is maximum

(b) the maximum distance.

2. A particle moves in a straight line with the velocity v = 2t – 4 where t is the time in

second after passing through point O. Find

(a) the displacement of the particle after 4 seconds

(b) the displacement when the particle stops momentarily.

3. A particle moves in a straight line with the velocity v ms-1

where v = 12 – 2t3

1 where t

is the time in seconds after passing through a fixed point O. Find

(a) the displacement of the particle after 4 seconds

(b) the maximum distance traveled by the particle before it changed its direction.



3. ACCELERATION

A. DETERMINE ACCELERATION FUNCTION OF A PARTICLE BY

DIFFERENTIATION

The instantaneous acceleration of a particle, a, at the instant t, is the rate of change

of velocity with respect to time, that is

a = dt

dv=

2

2

dt

sd

1. The uniform acceleration means that the velocity change in the unvarying

rate.

2. Meaning of the signs of acceleration:

0a velocity increases when t increases.

0a velocity decreases when t increases. ( deceleration or

retardation).

0a uniform velocity / v maximum or minimum

EXERCISE 9

A particle moves with its velocity v ms 1 and t is the time after passing through a fixed point

O. Find the initial acceleration and acceleration when t = 3 for each of the following.

Velocity formulae Initial acceleration Acceleration at 3 s

example:

v = 3t – t²

a= dt

dv= 3 –2t

When t = 0, a = 3–2(0)

= 3ms1

When t =3, a = 3–2(3)

= –3ms2

(a) v =2t² + 5t

(b) v = t³ + 2t²– 6



Displacement formulae Initial acceleration Acceleration at 3 s

(c) s = 2t³–t² + 8

(d) s = t – t³

(e) s = 4t – t³

B. DETERMINE VELOCITY FUNCTION OF A PARTICLE FROM

ACCELERATION FUNCTION BY INTEGRATION

EXAMPLE:

The acceleration of a particle which is moving along a straight line from its

instantaneous rest is a ms2 and t s is the time after passing through a fixed point O.

Find the maximum velocity of the particle.

Acceleration function Velocity function Maximum or minimum velocity

a = 6 – 2t v = a dt

= (6– 2t)dt

= 6t – t² + c

When t = 0, v = 0,

therefore c =0.

v = 6t – t²

Maximum velocity ; a = 0

6 – 2t = 0

t = 3

When t = 3,v = 6 (3) – 3² = 9 ms1



Acceleration function Velocity function Maximum or minimum velocity

(a) a = 6 – 4t

(b) a = 2t – 4

(c) a = 6t² –2t



EXERCISE 10

1. A particle is moving along a straight line and passed a fixed point O with its velocity,

v ms1. Given v = t(t –3) and t is the time in second after passed through O. Find

(a) displacement when t = 5 second

(b) maximum distance before it changed its direction

(c) total distance traveled in the first 5 second

2. A particle moves along a straight line through a fixed point O. Its acceleration, a cm s2,

is given by a = 2t + 3, where t is the time in seconds after passing through O. Given that

its initial velocity is –10 cm s1. Find

(a) the velocity of the particle when its acceleration is 11 cm s1

(b) the total distance traveled by the particle during 4 seconds after passing through 0.



3. A particle moves along a straight line passing through a fixed point O with velocity

1

2

1 ms . Its acceleration, a ms2 , is given by a = 3t – 2, where t is the time in seconds

after passing through O. Find

(a) the time at which the particle is at instantaneous rest

(b) the time at which the particle passes through O again.

MOTION ALONG A STRAIGHT LINE - Penditamuda's Blog · 2011-01-30 · Additional Mathematics Motion along a straight line [email protected] 3 B. DETERMINE THE TOTAL DISTANCE TRAVELLED

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