motionalongastraightlinemotionalon gastraightlinemotionalongastraightli nemotionalongastraightlinemotional ongastraightlinemotionalongastraigh

Size: px

Start display at page:

Download "motionalongastraightlinemotionalon gastraightlinemotionalongastraightli nemotionalongastraightlinemotional ongastraightlinemotionalongastraigh"

Noel Lawson
5 years ago
Views:

1 motionalongastraightlinemotionalon Additional Mathematics NR/GC/ Addmaths gastraightlinemotionalongastraightli nemotionalongastraightlinemotional ongastraightlinemotionalongastraigh MOTION ALONG A STRAIGHT LINE tlinemotionalongastraightlinemotion alongastraightlinemotionalongastraig Name htlinemotionalongastraightlinemotio... nalongastraightlinemotionalongastra ightlinemotionalongastraightlinemoti onalongastraightlinemotionalongastr aightlinemotionalongastraightlinemo tionalongastraightlinemotionalongast raightlinemotionalongastraightlinem otionalongastraightlinemotionalonga straightlinemotionalongastraightline motionalongastraightlinemotionalon gastraightlinemotionalongastraightli nemotionalongastraightlinemotional ongastraightlinemotionalongastraigh SMSJ/009 tlinemotionalongastraightlinemotion

2 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 DISPLACEMENT ORIENTATION 1.POSITIVE The particle is on the RIGHT of O..NEGATIVE 3.ZERO The particle is on the LEFT of O The particle is AT O or return to O again 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 -t s = 1² -(1) =-1 Meaning that the particle is 1m on the left of O s = 3² -(3) = 3 Meaning that the particle is 3m on the right O a) s = t² + (b) s = t² -t -1 (d) s = t³ - t² -3 zefry@sas.edu.my

3 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) = 8 When t = 4, s = 8 4(4)² = 8 64 = 56 t = 4 O t = 0 56 m 8 m Total distance traveled = = 64m zefry@sas.edu.my 3

4 EXERCISE 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 = t² 3 (b) s = 5 t² (c) s = 5t 1 (d) s = t 5t zefry@sas.edu.my 4

5 EXERCISE 3 1. A particle moves along the straight line with the displacement s m is given as s = t 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. A particle moves along the straight line with the displacement s m is given as s = t 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. 3. A particle moves along the straight line with the displacement s m is given as s = 60 t 5t 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 5 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 = 8t + t. Find the total distance traveled (a) in the first 5 seconds zefry@sas.edu.my 5

6 (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 t. Given that the particle moves to the right of O until t = seconds and then moves to the left towards O. Find the total distance traveled by the particle in the first 8 seconds. zefry@sas.edu.my 6

7 . 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 ds v = dt 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 3t v = 9 6t v = 9 6(3) = 9 t = 0, v = 4 (b) s= t² 4t + (c) s = t² 4t (c) s = t² -1t + 16 zefry@sas.edu.my 7

8 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 -1t Find (a) the time when the particle is at instantaneous rest (b) the range of t for the positive velocity Solution s = t² -1t + 16 v = ds = t 1 dt (a) If particle is at rest,v = 0, t 1 = 0 t = 6s (b) If positive velocity, then v > 0; t 1 > 0 t > 6s EXERCISE 5 1 A particle moves along a straight line with the velocity of v ms 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 (a) v = 9t² 4 Time when the particle comes instantaneously to rest the range of t when particle moves to the left (b) v = t² 3t (c) v = t + t 8 zefry@sas.edu.my 8

9 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=t³ 15t² + 36t, find (a) the displacement when the velocity is zero (b) time when the particle moved to the left.. 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 3t, 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 second. SOLUTION: s = vdt, s = ( 3t 4) dt = 3 t 4t, c c is a constant When t = 0, s = 0; 3(0) 0 = 4(0) c c = 0 3t Therefore s = 4t 3() when t =, s = 4() = 14 m zefry@sas.edu.my 9

10 EXERCISE 7 VELOCITY FUNCTION Displacement when t = 3 1. v = 5 + 3t. v = 4 t 8 3 EXERCISE 7: 1 A particle moves along a straight line with the velocity of v ms, 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 = ( 6 6t) dt v = 0 t = 0, s = 0 = 6t 3t + c 6 6t = 0 t = 1, s = 6(1) 3(1) = 3 t = 0, s = 0, c = 0 t = 3, s = 6(3) 3(3) = 19 t = 1 s = 6t 3t total distance = = v = 3t 5t v = t t zefry@sas.edu.my 10

11 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.. A particle moves in a straight line with the velocity v = t 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 1 where v = 1 t 3 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. where t zefry@sas.edu.my 11

12 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 dv d s a = = dt dt 1. The uniform acceleration means that the velocity change in the unvarying rate.. Meaning of the signs of acceleration: a 0 velocity increases when t increases. a 0 velocity decreases when t increases. ( deceleration or retardation). a 0 uniform velocity / v maximum or minimum EXERCISE 9 1 A particle moves with its velocity v ms 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 dv a= = 3 t dt When t = 0, a = 3 (0) When t =3, a = 3 (3) = 3ms 1 = 3ms (a) v =t² + 5t (b) v = t³ + t² 6 zefry@sas.edu.my 1

13 Displacement formulae Initial acceleration Acceleration at 3 s (c) s = t³ 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 ms 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 v = a = 6 t a dt Maximum velocity ; a = 0 6 t = 0 t = 3 = (6 t)dt = 6t t² + c When t = 3,v = 6 (3) 3² = 9 ms 1 When t = 0, v = 0, therefore c =0. v = 6t t² zefry@sas.edu.my 13

14 Acceleration function Velocity function Maximum or minimum velocity (a) a = 6 4t (b) a = t 4 (c) a = 6t² t zefry@sas.edu.my 14

15 EXERCISE A particle is moving along a straight line and passed a fixed point O with its velocity, v ms 1. 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. A particle moves along a straight line through a fixed point O. Its acceleration, a cm s, is given by a = t + 3, where t is the time in seconds after passing through O. Given that its initial velocity is 10 cm s 1. Find (a) the velocity of the particle when its acceleration is 11 cm s 1 (b) the total distance traveled by the particle during 4 seconds after passing through 0. zefry@sas.edu.my 15

16 3. A particle moves along a straight line passing through a fixed point O with velocity 1 1 ms. Its acceleration, a ms, is given by a = 3t, 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. zefry@sas.edu.my 16

CHAPTER 9 MOTION ALONG A STRAIGHT LINE FORM 5 PAPER 2

CHAPTER 9 MOTION ALONG A STRAIGHT LINE FORM 5 PAPER 2 PPER. particle moves in a straight line and passes through a fixed point O, with a velocity of m s. Its acceleration, a m s, t seconds after passing through O is given by a 8 4t. The particle stops after