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2. 6
F
6.1 p F F F
ˈ F F ˈ kg m/s
v
m p = mv
F 1
10 kg F 20 m/s F
p = mv
v = 20
F p = 10(20)
m ‫ ׵‬p = 200 kg m/s Ans
F 2
F 1 kg F F 5m F
p = mv ----1
F m v F v2 = u2 + 2 as
u=0
v2 = 0 + 2 (10)(5)
5.0 v = 10 m/s
F 1 F p = 1(10)
v
‫ ׵‬p = 10 kg m/s Ans
6.2
I F( )
ˈ F F F ˈ kg m/s
u v
F
m m I = FΔt = Δp = mv - mu
Δ t
F ˆ F www.schoolDD.com 1

3. F
. = ∆p
- F F t (I) Δp t
- F F ˈ
F ˈ +v +v -v -v
¬ F F F F F
F FF F F
F 3
F F 1 F F F F 2
u=0 Δp
= mv - mu ----1
t=2 F m, u, t v F v = u + at
v
= 0 + (10)(2)
v = 20 m/s
F 1 F Δp = 1(-20) 1(0)
‫ ׵‬Δp = -20 kg m/s Ans
F 4
F 50 F F 10 m/s F F F
F 10
u =10 m/s
Δt = 10
m
m
v =10 m/s
F F ˈ F ˈ
Δp
= mv - mu
= 0.05(-10) 0.05(10)
‫ ׵‬Δp = -1.0 kg m/s F Ans
F Δt = Δp
F F = Δp/Δt = -1.0/(10x10-3)
F ˆ F www.schoolDD.com 2

4. ‫ ׵‬F = -100 N F ( ) Ans
F 5
F 100 5 F 1.8 F
F F 2
. F
.
.
m u =0
v =0
5.0 t1 m
t2
1.8
uF
v
. Δp = ?
F ˈ ˈ
F Δp
= mv - mu F ----1 F m u F ,v
F uF
v2 = u2 + 2 as ( F )
u2 F = 0 +2 (10)(5)
u F = 10 m/s
v
v2 = u2 + 2 as ( )
0 = v2 +2 (-10)(1.8)
v = 6 m/s
F 1, Δp = mv - mu F
= 0.1(6) 0.1(-10)
‫ ׵‬Δp = 1.6 kg m/s Ans
. F=?
F FΔt = Δp ----1 F Δp Δt
F Δt F F F
F ˆ F www.schoolDD.com 3

5. F F t1 t2
F t1v = u + at F
10 = 0 + 10 t1 (v = u F = 10)
t1 = 1 s
t2 F v = u + at
0 = 6 + 10 t2 (u = v = 6)
t2 = 0.6 s
Δt = t (t1 + t2)
= 2 (1 + 0.6)
Δt = 0.4 s
F 1, FΔt = Δp
F = Δp/Δt = 1.6/0.4
‫׵‬F =4 N Ans
. N=?
mg
F
m
N
F ˈ F F
F = N mg
N = F + mg
= 4 + 0.1(10)
‫ ׵‬N = 5 N Ans
F 6
F F ˈ F F F F F F F .
F F ˈ 0.5 F 20 m /s
u =10 m/s
F F
N u =10 m/s v =0
f
mg
F ˆ F www.schoolDD.com 4

6. F F F F 1
F F F F F
F F F F F F F
FΔt = Δp = mv - mu
f Δt = mv mu
-µmg Δt = mv mu ( F f ˈ F )
Δt = (v u)/(-µg) = (0-20)/(-0.5(10))
‫ ׵‬Δt = 4 s Ans
¬ F = ma F 2 v = u + at F ..
F 7
1 FF 30 m/s F 0.2
F 40 m/s F
v = 40
mv - mu
ሬറ ሬറ
u = 30 ሬറ
-mu
F ሬറ
mv θ
ሬറ
mv
ሬറ
mu
ሬറ
Δp = mv - mu
ሬറ ሬറ
ˈ F FF F F F F
F . F F
F F F
Δp = ඥሺmvሻଶ ൅ ሺmuሻଶ
= ඥሺ1x40ሻଶ ൅ ሺ1x30ሻଶ
‫ ׵‬Δp = 50 kg m/s Ans
tanθ = mu/mv = 30/40 = 0.75
‫ ׵‬θ = 37° Ans
F Δt = Δp
F = Δp/Δt = 50/(0.2)
‫ ׵‬F = 250 N Δp Ans
F ˆ F www.schoolDD.com 5

7. F 8
0.2 F 10 m/s F F
F F ˈ F
u = 10 F (N)
F
400
v=?
t (ms)
1 2 3
Δp = mv mu ( F m, u Δp F v )F
. F F t Δp (F Δt = Δp)
-3
Δp = ½ (3x10 )(400)
Δp = 0.6 kg m/s ( ˈ F)
F Δp = mv mu
F F 0.6 = 0.2v 0.2(-10)
‫ ׵‬v = 13 m/s Ans
F 9
10 F . F ˈ 0.2 50 N F
10
u=0 F Δt = Δp = mv - mu ( F m, u, t v )
F
t = 10 v = ?
N F
(50 fk) Δt = mv - mu
50 N
(50 0.2(10)(10)) (10) = 10v 10(0)
fk 300 = 10v
mg
‫ ׵‬v = 30 kg m/s Ans
F 10
F F 10 F . F ˈ 0.5 0.3
F F ˈ F 10
F ˆ F www.schoolDD.com 6

8. F (N)
100
N
F
50
t (s)
f
mg 2 4 6 8 10
FF F .8 F F F F F F
F F
F F F F fs
F F
fs fs = µs N = µs mg
= 0.5 (10)(10)
fs = 50 N
F F ≥ fs F = 50 --> t = 1 s
F F Δt = 10 -1 = 9 s
F F Δt = Δp = mv mu ----1
F F = F - fk
F F Δt = F Δt - fk Δt
F Δt . F F t
F Δt = ½ (150x1) + ½ (100x14) ( Δt = 1-->10)
F Δt = 775 N s
fk Δt F fk Δt = (µk N) Δt = (µk mg) Δt
= (0.3x10x10)(9)
fk Δt = 270 N s
F F Δt = 775 270 = 505 N s
F 1 F 505 = 10v 10(0)
‫ ׵‬v = 50.5 m/s Ans
F ˆ F www.schoolDD.com 7

9. 6.3 F
F F
u1 u2 v1 v2
m1 m2 m1 m2 m1 m2
F
∑‫ ܘ‬F = ∑‫ܘ‬
m1u1 + m2u2 = m1v1 + m2v2
- F F
- F ˈ F ˈ F F
6.4
6.4.1 F
1. F = ( F )
∑‫ ܘ‬F = ∑‫ܘ‬
2. FF = F
∑ ۳‫ܓ‬ F
= ∑ ۳‫ܓ‬
F --> u1 + v1 = u2 + v2
F ˆ F www.schoolDD.com 8

10. F 11
1 F 10 m/s F 2 F 5m/s
F ˈ F
u1 = 10 u2 = 5 v1 = ? v2 = ?
m1 m2 m1 m2
F
F ˈ
ˈ F
F ∑p F = ∑p
m1u1 + m2u2 = m1v1 + m2v2
1(10) + 2(5) = 1(v1) + 2(v2)
20 = v1 + 2 v2 ----1
F u1 + v 1 = u 2 + v 2
10 + v1 = 5 + v2
5 = v2 - v1 ----2
2 2 F
1+2, 3 v2 = 25
‫ ׵‬v2 = 8.33 m/s ( ˈ ) Ans
2, v1 = v2 5 = 8.33 - 5
‫ ׵‬v1 = 3.33 m/s Ans
F 12
F 11
.
. 1 F 2 F
. v1, v2 = ?
u1 = 10 u2 = 5 v1 = ? v2 = ?
m1 m2 m1 m2
F
F ˆ F www.schoolDD.com 9

11. F ˈ
F ∑p F = ∑p
m1u1 + m2u2 = m1v1 + m2v2
1(10) + 2(-5) = 1(v1) + 2(v2)
0 = v1 + 2 v2 ----1
F u1 + v 1 = u 2 + v 2
10 + v1 = -5 + v2
15 = v2 - v1 ----2
2 2 F
1+2, 3 v2 = 15
‫ ׵‬v2 = 5.0 m/s ( ˈ ) Ans
2, v1 = v2 15 = 5 - 15
‫ ׵‬v1 = -10 m/s F ( ˈ ) Ans
. u2 = 0 , v1, v2 = ?
u1 = 10 u2 = 0 v1 = ? v2 = ?
m1 m2 m1 m2
F
F ˈ
F ∑p F = ∑p
m1u1 + m2u2 = m1v1 + m2v2
1(10) + 2(0) = 1(v1) + 2(v2)
10 = v1 + 2 v2 ----1
F u1 + v 1 = u 2 + v 2
10 + v1 = 0 + v2
10 = v2 - v1 ----2
2 2 F
1+2, 3 v2 = 20
‫ ׵‬v2 = 6.67 m/s ( ˈ ) Ans
2, v1 = v2 10 = 6.67 - 10
‫ ׵‬v1 = -3.33 m/s F ( ˈ ) Ans
F ˆ F www.schoolDD.com 10

12. 6.4.2 F F
1. F =
∑‫ ܘ‬F = ∑‫ܘ‬
2. FF ≠ F
∑ ۳‫ܓ‬ F
≠ ∑ ۳‫ܓ‬
- F ˈ ˈ
- F F ˈ F F
F 13
m1 = 2 kg F 12 m/s F m2 = 4 kg F F
F F F F ˈ F F
u1 = 12 u2 = 0 v12 = ?
m1 m2 m1 m2
F
F ˈ
F ∑p F= ∑p
m1u1 + m2u2 = m1v1 + m2v2 = (m1+m2) v12
2(12) + 4(0) = (2+4) v12
‫ ׵‬v12 = 4.0 m/s Ans
FF ∑E F = ½ m1u12 + ½ m2u22
= ½ (2)(122) + 0
∑E F = 144 J
F ∑E = ½ m1v12 + ½ m2v22 = ½ (m1+ m2) v122
= ½ (2+4)(42)
∑E = 48 J
F = ∑E - ∑E F
F ˆ F www.schoolDD.com 11

13. = 48 144
= - 96 J Ans
∑E F ≠ ∑E ‫ˈ ׵‬ F F Ans
F 14
F 1 F 2 m/s 60 ° F 4
F 1 m/s F F
u1 = 2
60° u2 = 1 v12 = ?
m1
m2
F
F ˈ
F ∑p F = ∑p
m1u1 + m2u2 = m1v1 + m2v2 = (m1+m2) v12
m1u1cos60+ m2u2 = (m1+m2) v12
1(2 )(1/2)) + 4(-1) = (1+4) v12
-3 = 5 v12
‫ ׵‬v12 = - 0.6 m/s F Ans
F 15
ˀ 50 F 400 m/s F F F 2 kg F ˀ
F F F F F 1 m/s ˀ F F ˈ F
u1 = 400 u2 = 0 v2 = 1
v1 = ?
m1 m2 m2
F
F ˈ
F ∑p F = ∑p
m1u1 + m2u2 = m1v1 + m2v2
0.05(400) + 2(0) = 0.05 v1 + 2(1)
18 = 0.05 v1
‫ ׵‬v1 = 360 m/s Ans
F ˆ F www.schoolDD.com 12

14. F 16
ˀ F ˀ F ˈ 800 m/s ˀ
ˀ ˈ 20 g 1000 g ˀ F
u1 = 0 v2 = ? v1 = 800
m1
m1
u2 = 0 m2 m2
F
ˀ ˈ F F
F ˈ
F ∑p F = ∑p
m1u1 + m2u2 = m1v1 + m2v2
0.02(0) + 1(0) = 0.02(800) + 1(v2)
‫ ׵‬v2 = - 16 m/s F Ans
FF ∑E F = ½ m1u12 + ½ m2u22
=0 (u1 = u2 = 0)
∑E F =0 J
F ∑E = ½ m1v12 + ½ m2v22
= ½ (0.02)(8002) + ½ (1)(-16)2
∑E = 6528 J
F = ∑E - ∑E F
= 6528 0
= 6528 J Ans
F 17
2 1 kg 2 kg F F F F F
1 kg F 2 m/s 2 kg F F
u1 = 0 u2 = 0 v1 = 2 v2 = ?
m1 m2 m1 m2
F F F
F ˆ F www.schoolDD.com 13

15. F ˈ
F ∑p F = ∑p F F
m1u1 + m2u2 = m1v1 + m2v2
1(0) + 2(0) = 1(-2) + 2(v2)
‫ ׵‬v2 = 1 m/s Ans
F 18
A 40 kg F F F F 2 m/s
F F 100 kg
v1 = 2
u1 = 0
v2 = ?
u2 = 0 m1 m1
m2 m2
F
F ˈ
F ∑p F = ∑p
m1u1 + m2u2 = m1v1 + m2v2
40(0) + 100(0) = 40(2) + 100(v2)
‫ ׵‬v2 = - 0.8 m/s F Ans
F 19
50 kg F F 5 kg F F F 2 m/s F
F F F F F F
u1 = u2 = 2 v1 = ?
m1 v2 = 0 m1
m2 m2
F
F ˈ
F ∑p F = ∑p
m1u1 + m2u2 = m1v1 + m2v2
F ˆ F www.schoolDD.com 14

16. 50(2) + 5(2) = 50(v1) + 5(0) ( )
110 = 50 v1
‫ ׵‬v1 = 2.2 m/s ( ) Ans
F F F
F 20
ˀ 10 g F 400 m/s F ˆ F F 2 kg F F ˀ F .
ˈ 0.2 F F F F
v12
u1 = 400 u2 = 0 v=0
m1+ m2
m1 m2 f
A s B
F
F = ∑p
∑p F
m1u1 + m2u2 = m1v1 + m2v2 = (m1 + m2) v12
0.01(400) + 2(0) = (0.01 + 2) v12
4 = 2.01 v12
‫ ׵‬v12 = 2 m/s
F F ˀ ˀ F s F F
F A F F A B
A --> B
F - WA ◊ B ∑ E୅ = ∑ E୆
½ (m1 + m2) v122 - f s = 0
½ (m1 + m2) v122 - µ(m1 + m2)g s = 0
s = ½ v122/ (µg)
= ½ (22)/ (0.2x10)
‫ ׵‬s = 1.0 m Ans
F 21
ˀ 20 g ˂ 1 kg ˆ F ˂ F F 0.9 m
F ˂ F F 60 ˀ F ˂
60°
v=0
l = 0.9
u1 = ? h
m2
m1 u2 = 0
F ˆ F www.schoolDD.com 15

17. F ∑p F = ∑p
m1u1 + m2u2 = m1v1 + m2v2 = (m1 + m2) v12 (v12 = ˂ ˀ )
0.02(u1) + 1(0) = (0.02 + 1) v12
0.02 u1 = 1.02 v12 ----1
F ∑ Eଵ = ∑ Eଶ
½ (m1 + m2) v122 = (m1 + m2)gh
v122 = 2gh = 2g(l l cos60)
v122 = 2(10)(0.9-0.9(1/2))
‫ ׵‬v12 = √9 = 3.0 m/s
F v12 1, 0.02 u1 = 1.02 (3)
‫ ׵‬u1 = 153 m/s Ans
F 22
2 kg 6 kg F 10 m/s 4 m/s
F F F k = 3400 N/m
u1 = 10 u2 = 4 v12
m1 m2 m1 m2
F
F ∑p F = ∑p
m1u1 + m2u2 = m1v1 + m2v2 = (m1 + m2) v12 (v12 = )
2(10) + 6(4) = (2 + 6) v12
v12 = 5.5 m/s
FF ∑E F = ½ m1u12 + ½ m2u22
= ½ (2)(102) + ½ (6)(42)
∑E F = 148 J
F ∑E = ½ (m1 + m2) v122
= ½ (2 + 6)( 5.52)
∑E = 121 J
FF F F ˈ F
F
F ˆ F www.schoolDD.com 16

18. ∑E F = ∑ E + Ep
148 = 121 + ½ kx2
x2 = 0.01
‫ ׵‬x = 0.1 m Ans
F 23
F 1000 kg F 5 m/s F 2000 kg F
F F 0.2 m/s F F F F
. F F ˈ 0.26
u2 = 0 v2 = ?
u1 = 5 v1 = 0.2
m1 m2 m1 m2
F
F = ∑p
∑p F
m1u1 + m2u2 = m1v1 + m2v2
1000(5) + 2000(0) = 1000(-0.2) + 2000(v2)
v2 = 2.6 m/s
vA = 2.6 vB = 0
m m
f
A s B
A --> B F A F
A --> B F B
F ˆ F www.schoolDD.com 17

19. F ∑ E୅ - WA ◊ B = ∑ E୆
½ mvA2 f s = 0
½ mvA2 µmg s = 0
s = ½ (2.6)2/(0.26x10)
‫ ׵‬s = 1.3 m Ans
F 24
F 5 kg 5m F F F 15 kg
F . F 0.5 F F ˈ
F
uA = 0
m1 A
u1 = ? v12 v=0
5 u2 = 0
m1 m1 m1
m2 m2 f m2
B B s C
F
m1 F m2 F
F ∑ E୅ = ∑ E୆
mgh = ½ mu12
u12 = 2gh = 2(10)(5)
u1 = 10 m/s
F = ∑p ∑p F
m1u1 + m2u2 = m1v1 + m2v2 = (m1 + m2) v12
5(10) + 15(0) = (5 + 15) v12
v12 = 2.5 m/s
ˀ F s
B --> C F B F
B --> C F C
F ∑ E୆ - WB ◊ C = ∑ Eେ
½ mvB2 f s = 0
½ (m1 + m2) v122 µ(m1 + m2)g s = 0
s = ½ (2.5)2/(0.5x10)
F ˆ F www.schoolDD.com 18

20. ‫ ׵‬s = 0.625 m Ans
6.5 2
ˈ FF F (cm.) F F F
v2
u2
u1 m2
cm. +
m1
+ m2 m1
+ m2
m1 v1
F
F F F
∑‫ ܘ‬F = ∑‫ܘ‬
F ˈ F
∑ ۳‫ܓ‬ F
= ∑ ۳‫ܓ‬
- F( F )
x y F F
∑ p୶ F = ∑ p୶
∑ p୷ F = ∑ p୷
- F ˈ F F F
F 25
A m F 10 m/s F B 2m FF F
F A 30° B 60° A F
A B ˈ F
F ˆ F www.schoolDD.com 19

21. vA
uA= 10
uB = 0 mA 30°
mA
mB 60°
mB
F vB
ˈ
F ∑p F = ∑p
0
mAሬuA + mBሬuB = mAvA + mBvB
റ റ ሬറ ሬറ
mAሬuA = mAvA + mBvB
റ ሬറ ሬറ
1 F
mAvA
ሬറ 60°
mAuA
ሬറ
mBvB
ሬറ
30°
mAvA
ሬറ
mBvB
ሬറ
30° mAuA
ሬറ
60°
F mAuA
F
mAuA cos30 = mAvA
vA = uA cos30
= 10 (√3/2)
‫ ׵‬vA = 5 √3 m/s Ans
mAuA sin30 = mBvB
vB = (mA/ mB) uA sin30
= (1/2) (10)(1/2)
‫ ׵‬vB = 2.5 m/s Ans
2 x y F x y
0
x ∑ p୶ F = mAuA + mBuB
= m(10)
F ˆ F www.schoolDD.com 20

22. ∑ p୶ = mAvA cos30 + mBvB cos60
= m vA(√3/2) + 2m vB (1/2)
= (√3/2) m vA + m vB
F ∑ p୶ F = ∑ p୶
m(10) = (√3/2) m vA + m vB
10 = (√3/2) vA + vB ----1
y ∑ p୷ F = 0
∑ p௬ = mAvA sin30 mBvB sin60
= m vA(1/2) - 2m vB (√3/2)
= (1/2) m vA - √3m vB
F ∑ p௬ F = ∑ p୷
0 = (1/2) m vA - √3m vB
vA = 2 √3 vB
F vA 1, 10 = (√3/2) (2 √3 vB) + vB
vB = 10/4
‫ ׵‬vB = 2.5 m/s Ans
v1 = 2 √3 v2
v1 = 2 √3 (2.5)
‫ ׵‬v1 = 5 √3 m/s Ans
F 26
F 1500 kg F F 20 m/s ʽˋ 2000 kg ˁ
F F F F
v12 = ?
u1= 20
m1
m1
m2
m2 u2= 20
F
F ∑p F = ∑p
m1ሬu1 + m2ሬu2
റ റ = m1v1 + m2v2
ሬറ ሬറ
m1ሬu1 + m2ሬu2
റ റ = (m1+ m2) v12
ሬറ
F ˆ F www.schoolDD.com 21

23. F
(m1+ m2) v12
ሬറ
α
m2u2
ሬറ
m1u1
ሬറ
F m1u1 m2u2
(m1+ m2) v12 F
((m1+ m2) v12)2 = (m1u1)2 + (m2u2)2
((1500+ 2000) v12)2 = (1500x20)2 + (2000x20)2
v12 = (5 x 104)/3500
‫ ׵‬v12 = 14.2 m/s Ans
tan α = m2u2/ m1u1 = 2000/1500 = 4/3
‫ ׵‬α = 53° Ans
F 27
F F 2 ˈ F F F
F 30 m/s F F F
m u=0
v1 = 30
t=2 m/2
m/2 v2 = ?
m v = u1 = ?
F
F u1 t=2s
v = u + at
u1 = 0 + 10(2)
u1 = 20 m/s
F ∑p F = ∑p
mሬu1 = (m/2) v1 + (m/2) v2
റ ሬറ ሬറ
F ˆ F www.schoolDD.com 22

24. F
(m/2) v1
ሬറ
α
mu1
ሬറ
(m/2) v2
ሬറ
F u1 v1
v2 F
(v2/2)2 = (u1)2 + (v1/2)2
(v2/2)2 = (20)2 + (30/2)2
‫ ׵‬v2 = 50 m/s Ans
tan α = mu1/ ((m/2)v1) = 2(20)/30 = 4/3
‫ ׵‬α = 53° Ans
F 28
F F F F ˈ 3 F F F 60°
10 m/s F ˈ F F F F
F ˈ F
v1 = 10
m 60° v2 = 10
u=0 v3 = ? m
M
2m
F
F ∑p F = ∑p
Mሬu = m v1 + m v2 + 2m v3
റ ሬറ ሬറ ሬറ
0 = m v1 + m v2 + 2m v3
ሬറ ሬറ ሬറ
F
F F
- 2mv3
ሬറ
mv1
ሬറ F F F
60° F F ∑p = 0 = ∑p F
2mv3
ሬറ mv2
ሬറ
F ˆ F www.schoolDD.com 23

25. F v1 v2
v3 F
R = √Aଶ ൅ Bଶ ൅ 2ABcosθ
2v3 = ඥvଶ ൅ vଵ ൅ 2vଶ vଵcos60
ଶ ଶ
2v3 = ට10ଶ ൅ 10ଶ ൅ 2ሺ10ሻሺ10ሻሺଵሻ
ଶ
2v3 = √300
‫ ׵‬v3 = 5 √3 m/s Ans
“ถาอยากเปนจอมยุทธ ผูชนะสิบทิศ ก็ใหหมันฝกฝน
 ่
ทบทวนตํารา แตถาพอใจเปนแคทหารเลวปลายแถว ก็
ถ พอใจเป ทหารเลวปลายแถว
ทหารเลว
จงขี้
จงขีเกียจตอไปเถิด”
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