1. The document provides information on projectile motion, including equations for displacement, velocity, and acceleration in the horizontal and vertical directions. 2. Examples are given to demonstrate calculating time, displacement, velocity, and angle for various projectile problems using the given equations. 3. Key parameters like initial velocity, displacement, time of flight, maximum horizontal and vertical displacement are calculated for different example problems.

1. SchoolDD.com
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2. 4
F F F
F F F
F F F
4.1 F
F (projectile) F F F F F
F F F
F 2 F
F F F F
F F 2
1. F F F
y
vy = 0
vy
vx = ux
u vx = ux vx = ux
uy = u sin
vy vx = ux
ux = u cos
x
vy
- F
- F F F4 ( a=
g)
- F F sx = uxt
- F
2. F
y
u ux =u
uy = 0 vx = ux
vy vx = ux
vy
vx = ux
x
vy
– F F www.schoolDD.com 1

3. - F
- F F F F F g
- F F sx = uxt
2 F
t F F F
F F F F F F F
F F F F F F F
...
F 4 F F ...
1. v = u + at
2. s = (u + v)/2 x t
3. s = ut + ½ at2
4. v2 = u2 + 2as
F a F g +, - F u +, F u ..
F F
F F F F
y
u
uy h
θ ux
x
sx
F ux = u cos uy = u sin
F t
F
sy = uy t + ½ ay t2
sy = 0 ay = g F u
F F 0 = u sin t + ½ (-g) t2
t = (2 u sin )/g ---(1) ( F g = 10 , t = uy/5)
F h
vy2 = uy2 + 2ays
– F F www.schoolDD.com 2

4. vy = 0
F F 0 = (u sin )2 + 2(-g)h
h = (u sin )2/ 2g ---(2)
F F t h F F u sin = (2) (1)
F F t = 2( )/g
t=
FF F F F F F F F
F F F F F F
F sx
sx = uxt
F t = (2 u sin )/g (1) F sx = (u cos ) (2 u sin )/g
sx = u2 (2 sin cos /g ---(3)
sx = u2 (sin ) /g ---(4)
(3) (4) F F ...
1. F u F sx sin F sin = 1.0 = 90° = 45°
2. F sx F u F = 45° F F
3. F u F sx F F F = 90 F 30 60
F sin30 cos30 = sin60 cos60 , 37° 53° F sin37 cos37 = sin53 cos53
F .....
60 60
45 45 u F
u F 30 30
u F F
F F F F ?
F F F F ?
F
F F F F ?
– F F www.schoolDD.com 3

5. F
- F F F
- F F F g
- F F F ( + )
- F F
- F
F F
- F F F
F F
- F F
F F F F ..
F 1
F F F 20 m/s 30
. F
. F
.
. F F 1.5
. F 1.5
F F F F F FFF
y “ u
u = 20 m/s F F
uy = u sin30 sy F F”
30 ux = u cos30
x
sx
. F t
F F
vy = 0 vy = 0
v = u+a t
- ay
v = u + (−g) t
y y
+ uy - vy
0 = usin30 − g t
– F F www.schoolDD.com 4

6. 1
usin30 20 × 2
t= =
g 10
∴ t = 1 s Ans
F F F ?
. F sy max = ?
F . F F t=1s vy = 0
u+v
s=  t
 2 
 u y + vy 
sy = 
 2 t 
 
20 × 1
sy max =  usin 30 + 0  t =
 
2 ×1
 2  2
= 5.0
∴ F sy max = 5.0 m Ans
. sx max = ?
F sx = uxt ( ux )
F = 1x2 = 2
sx max = ucos 30 × t
3
= 20 × ×2
2
∴ sx max = 20 × 3 m Ans
. F F t = 1.5 s
t = 1.5 s
1 2
sy F s = ut + at
2
1
sy = u y t + (-g)t 2
2
= (usin 30 )t − 1 gt 2
2
= (20 × 1 2 )× 1.5 − 1 × 10(1.5) 2
2
= 15 11.25
sy = 3.75 m
sx F sx = uxt
= ucos 30 × t
– F F www.schoolDD.com 5

7. 3
= 20 × × 1.5
2
sx = 15 3 m
∴ F F F 15 3 m F 3.75 m F Ans
. F v=? t = 1.5 s
vy F v = u + at
vy = usin 30 + (−g)t
vy = 20 × 1 − 10 × 1.5
2
vy = -5 m/s
( ∴ F u )
vx = ux = ucos 30
vx = 20 ×
2
3
vx = 10 3 m/s
F v F F F
v = v2 + v2
x y ( vx vy )
F
vx
α
= (10 3 ) 2 + (5) 2 = 325
vy v v = 18 m/s
ν
y 5 1 3
tan −1 α = = = =
ν 10 3 2 3 6
x
3
α = tan −1
6
∴ F 18 m/s tan −1
3
Ans
6
F 2
10 F 20 /
. F F
. F F F F
. F
– F F www.schoolDD.com 6

8. u = 20 m/s “ F F F
ux = u = 20 m/s FF FF F ”
uy = 0
sy = 10 m
vx = ux
sx vy
. t=?
F F F F
1 2
s = ut + at
+ uy
2 + sy
1 2
sy = 0+ gt + ay = g
2
1
10 = (10)t 2 + vy
2
t = 2 s
∴ F t= 2 s F Ans
. sx max = ?
F
sx = uxt
sx max = 20 × 2 (t = 2 F .)
∴ sx max = 20 2 m Ans
. v=?
u + v
s =  t
 2 
 uy + vy 
sy = 
 2 t

 
 0 + vy 
10 = 
 2  2
 (t = 2 F .)
 
vy = 10 2 m/s
vx = ux = u = 20 m/s
– F F www.schoolDD.com 7

9. ∴ v = vy + vx
2 2
vx
α
= (10 2 ) 2 + 20 2 = 600
vy v v = 10 6 m/s
vy 10 2
tan α = =
vx 20
2
α = tan −1
2
∴ 10 6 m/s tan −1
2
Ans
2
F 3
F F F F F 2,000
F 200 F F F F
4,000 F F
u = 200 m/s
ux = u = 200 m/s “ F F F F
uy = 0 F FF FF F F
F FF F F F sx max F
sy = 2000 m
4000 . F
sx max =uxt, F ux t F .... ”
sx max
F
1 2
s = ut + at
2
1
sy = u y t + (g)t 2
2
1
2000 = 0 + (10)t 2
2
t = 20 s
ux = u = 200 m/s
sx = u x t
sx max = 200 (20) = 4000 m
sx max F ∴ Ans
– F F www.schoolDD.com 8

10. F 4
F F F 5 F F F F F F F
F F F
u = ux = ?
“ F . F . F
sx max = uxt, F sx max t F .... F ”
uy = 0
5.0 m
5.0 m
F
s = ut + 1 at 2
2
1
sy = u y t + (g)t 2
2
5 = 0 + 1 (10)t 2
2
t = 1 s
sx = u x t
5 = u x1
ux = 5 m/s
∴ F F F F 5 m/s !! Ans
F 5
20 F 37 F
25 / F F FF F F F 15 F
F
ux
37° u = 25 m/s
uy “ F . F 3. F uy ≠ 0”
sy = 20 m
15.0 m
– F F www.schoolDD.com 9

11. F F
ux = ucos37 o = 25x  4  = 20 m/s
 
5
uy = usin37 o = 25x  3  =
  15 m/s
5
F
1 2
s = ut + at
2
1
sy = u y t + (g)t 2
2
20 = 15t + 1 (10)t 2
2
= 0
t 2 + 3t − 4
(t-1) (t+4) = 0
∴t =1 s
sx max sx = u x t
sx max = 20 x 1 = 20 m 15 m
∴ F Ans
F 6
F F 5.00 F F
F F F F F F
uy = u sin45 “F F .. F F
u=?
”
45 ux = u cos45
sx max = 5.0 m
F F F F
F F 45o
F F F
ux = ucos45 o
uy = usin45 o
ux = uy
( F F )
F
1 2
s = ut + at
2
– F F www.schoolDD.com 10

12. 1
sy = u yt + a yt2
2
1
0 = u y t + (-10)t 2
2
uy
t = = ux
5 5
sx = uxt
u 
5.0 = ux  x 
 5 
ux = 5.0 m/s
ux 5
u = = 1
cos45
2
u = 5 2
∴ F F 5 2 m/s Ans
F F F F F F F 45o F
F 60o F F F F 5.00
F 7
F F 60 53 o F 25 /
. F F
. F F F
. F F F
u = 25 m/s
uy “ F F F
53°
FF FF F ”
ux
sy = 60 m
sx max
– F F www.schoolDD.com 11

13. . F t= ?
 3
ux = u cos53 = 25   = 15 m/s
5
4
uy = u sin53 = 25   = 20 m/s
5
( F F
F )
1 2
s = ut + at vy = 0
2
1
sy = u yt + a yt2 + uy
2 + sy
-60 = 20t + ½(-10)t2 - ay
- sy
t2 4t -12 = 0 u=
- vy
(t-6) (t+2) = 0
∴ t = 6 s Ans
. sx max = ?
sx = u x t
sx max = u x t
= 15 x 6
∴ sx max = 90 m Ans
. sy max = ?
F
vy= 0
v2 = u 2 + 2as
v2
y = u 2 + 2a y s
y
0 = 202 + 2 (-10) sy
sy = 20 m
sy max = 60 + sy = 60 + 20
∴ = 80 m Ans
– F F www.schoolDD.com 12

14. 4.2
F F F
v
v2
Fc = mac ac =
m r
T = Fc 2πr
v2 v = ωr = = 2πrf
= m T
r
θ 2π
ω = = = 2πf
t T
s
“ T F θ = r s
r θ
2
F F F T = Fc = mv /r
F v F F F F F
F F F
F F F F F F Fc F m
F F F ac
F F F F
“ F F F r F v
v
T F1 F
r θ 1
f F 1 F (f= )
T
F 1/ F (Hz)
F v F1 (= 2πr ) F 1 (=T)
2πr
(v= ) F / 2
T
ω F1 (= 2π ) F 1 (= T)
2π
(ω = ) F /
T
“ F F F F F”
F F
F F F F
F F F F F F
F 4
T f ω
– F F www.schoolDD.com 13

15. F 8
F 0.1 1.0 F 10 /
.
.
. F
.
. F F F
. F F
r =1.0 m
v = 10 m/s “ F F F
m = 0.1 kg FF FF F ”
T = Fc
. T=?
F v, r T F
2πr
v =
T
2πr 22
T = = 2× × 1.0
v 7 10
∴ T = 0.63 s. Ans.
. f=?
1
f =
T
1
=
0.63
∴f = 1.59 Hz Ans.
. F
∴ v = 10 m/s Ans.
. =?
v = ωr
v 10
ω = =
r 1.0
∴ ω = 1.0 rad/s Ans.
– F F www.schoolDD.com 14

16. . F F F ac =?
2
v
ac =
r
102
=
1.0
∴ ac = 100 m/s2 Ans.
. F F Fc =?
Fc = mac
= 0.1 × 100
∴ Fc = 10 N Ans. ( F F )
F 9
F F 1
30 F F F 2 F
F F
l=1m
“ F F F F
T
v Tsin30 FF F T
30°
2 ”
Tcos30
mg “ T cos30 F F F
w=2N
T=W=2N F F F F F
Tcos30 = Fc F F
 3
∴ Fc = 2


 = 3 N Ans.
 2 
Tsin30 = mg
2 1
m = ×  = 0.10 kg
10  2 
ac Fc = mac
ac = 3 / 0.10
∴ ac = 10 3 m/s2 Ans.
2
v
v ac = , r= l cos 30
r
– F F www.schoolDD.com 15

17. 3
v2 = 10 3 (1.0 × )
2
∴ v = 15 m/s Ans.
F 10
F F F
37o
“ F F F F
FF F T
2 ”
37° 37°
l=1m T
Tcos37
37°
Tsin37
m
mg
“ T sin37 F F F
Tsin37 = Fc
mv 2
Tsin37 = ---------(1)
r
∑F y =0
Tcos37 = mg ----------(2)
v2
(1)/(2) , tan37 = ( r = l sin37 = 1x 3 )
rg 5
3 3
v2 = × × 10
4 5
3
∴ v = m/s Ans.
2
ω v = ωr
3 5
ω = ×
2 3
5
∴ ω = rad/s Ans.
2
– F F www.schoolDD.com 16

18. F 11
F 0.4 3 F F F F F 3 F F F 1
F F F F F F
12 /
“ F F F F
FF F ”
v3 = 12 m/s
T3 T3 T2 T2 T1
m3 m2 m1
3 F F F
T, f ω F
ω F 3 v = ωr
12 = ω (3)
ω = 4 rad/s
T3 m3
T3 = Fc
2
m3v3
v3 T3 =
r
T3 0.4 × (12) 2
=
m3 3.0
T3 = 19.2 N Ans.
T2 m2
v2 T2 T3 = Fc ( F F F F )
T3 T2 T2 T3 = m2 ω2 r
m2 T2 19.2 = 0.4 × 42 × 2
∴ T2 = 32 N Ans.
T1 m1
T1 T2 = Fc
v1
T1 T2 = m1 ω2 r
T2 T1
T1 32 = 0.4 × 42 × 1
m1 ∴ T1 = 38.4 N Ans.
– F F www.schoolDD.com 17

19. 4.3 F
1. F
F F F F F F F
. F fs = Fc
0 < f s ≤ µs N
“ fs F F 0 < Fc ≤ µ s N
F F mv 2
0< ≤ µ s mg
r
fs r
fs = Fc 0< v2
≤ µs
rg
s
v 0< v ≤ µ s rg
“ F ”
“ 0< v ≤ µ s rg F F Fv F F
F F F F F0 F µ s rg ”
. F
F F F F F FR N
fs F F
Rcosθ = N
θ
R
R
tan θ = ν2
rg
cm θ
Rsinθ = fs
cm
mv 2
N mg R sin θ = Fc =
r
fs mg R cos θ = mg
fs
tan θ =
“ ” “ R sinθ F N
F F 0 < f s ≤ µs N
F 0 < tan θ ≤ µ s
tan θ = ν v
2
“ F F θ F F
rg
F F F F F F
F F F ... .. F F ..
F F F F F tan θ ≤ µ s ”
– F F www.schoolDD.com 18

20. 2. F
F F F F F
. F
“ N sinθ F F
Ncosθ
θ
N Nsinθ = Fc F ( F F F
)
Nsinθ
mg θ mv 2
N sin θ = Fc =
r
N cos θ = mg
tan θ=ν v
2
“ F F θ F F v2
rg tan θ =
F F F F F θ F rg
F v F F F ... F F F F ..
. F
Ncosθ
θ N
Nsinθ
tan θ = ν2
rg
mg θ
tan θ = ν
2
“ F F θ F F
rg
v θ F F F F ... F
F F F ..
– F F www.schoolDD.com 19

21. F 12
F FF 72 / F 100 400
F F . F F 0.1
“ F F F F
fs F F ”
fs r
= 0.1
v = 20 m/s
103
v = 72 ./ . = 72 × / = 20 m/s
60 × 60
fs = Fc
mv 2
µN =
r
mv 2
µmg =
r
v = µrg F F F F
r = 100 m, v = 0.1(100) ×10 = 10 m/s < 20 m/s F Ans
r = 400 m, v = 0.1(400) ×10 = 20 m/s ≥ 20 m/s Ans
F 13
F F F F F F .
F F F
fs = fc F
mv 2
µmg =
r
2
v
=
rg
F v2 r g
∴ =
F F
=
µ
∴ = Ans
4
– F F www.schoolDD.com 20

22. F 14
F F F 15 / F 30 F
F
Rcosθ
θ R
Rsinθ
N mg
fs
v2
tan θ =
rg
152 15 3
tan θ = = =
30 × 10 20 4
∴ θ = 37o
∴ F 90 - θ = 53o Ans
F 15
F F F 200 tan
θ = 0.45 F F F F F F
F F F F F F
Ncosθ “ F F F FF
N
FF F ”
θ
fs Nsinθ
fscosθ
fssinθ mg θ
fs = 0
v2
tan θ =
rg
v2
0.45 =
200 (10 )
v = 30 m/s = 108 km/hr
∴ F F F 108 km/hr Ans
– F F www.schoolDD.com 21

23. F 16
F F F F F F F F F 9 .
F F µ s = 0.1 F F F F F F F F
F ( F cm. F F 0.5 )
r
fs
N θ
0.5
mg
N = Fc ( F F F F F )
mv 2
N = -----------(1)
( r − 0 . 5)
∑F = y 0
fs = mg
µs N = mg ------------(2)
1 v2
(1)/(2) , =
µs ( r − 0 . 5) g
1 v2
=
0 .1 (4.5 − 0.5)g
∴ v = 20 m/s = 72 km/hr
fs µs N
tan θ = = = µs = 0.1
N N
∴ F F F F 72 km/hr tan-1 0.1 F F Ans
– F F www.schoolDD.com 22

24. 4.4
F F F
F F F F
T mg ( F F F F F ) F
F F FC F r F v
v D
mg mν 2
A TA mg =
TD v r
mν 2
TC
C
B TB mg cos θ=
r
θ TB mg mν 2
TA
v C TC =
B r
TD + mg = mν
2
A v
θ
mg cosθ D
r
mg
mg
F
- F F F F F
F F F F
F .F . F F F F F F
F F
- F F F F 4 F
- v F F F F F
v F A F D
- F F D F F
0 mv 2
F T=0 F TD + mg = --> v = rg
r
v
1
v ≥ rg 1
2 3 v < rg 2
v =0 3
– F F www.schoolDD.com 23

25. F 17
1 1 F F F F
F F F F F F F A,
B,C, D
mg cos60
D D
C mg C
TC
TD
60° mg
60°
B TB B
TA mg
A
A
mg
“ F F F FF
FF F ”
v F F F D
F ∴ TD = 0 Ans
D TD + mg = Fc
mv 2
TD + mg =
r
mv 2
0 + mg =
r
v2 = rg = 1.0 x 10 = 10
A TA mg = Fc
mv 2
TA mg =
r
mv 2
TA = + mg
r
1(10 )
F v 2 = 10 F TA = + 1(10 )
1
∴ TA = 20 N Ans
B TB = Fc
mv 2
TB =
r
1(10 )
=
1
– F F www.schoolDD.com 24

26. ∴ TB = 10 N Ans
C TC + mg cos 60 = Fc
mv 2
TC + mg cos 60 =
r
mv 2
TC = - mg cos 60
r
1(10)
= - 1(10) ( 1 )
1 2
∴ TC = 5N Ans
F 18
F F F 8 F F F
F F 1 4 F
N = ¼ mg
v=?
“ F F F FF
mg FF F ”
r=8m
N + mg = Fc + F F
mv 2
N + mg =
r
mg mv 2
+ mg =
4 r
5
v2 = rg
4
5
= (8)(10)
4
∴ v = 10 m/s Ans
F
- FF
F F F F F F F F
mv 2
FC = F F F F
r
– F F www.schoolDD.com 25