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Physics
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Session
Kinematics - 3
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Session Objectives
Problems ?
Free fall under gravity
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Free Fall Under Gravity
- g
g is constant = - g
-y
Motion under gravity is along y-axis. Downward is negative
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One dimensional equations of motion under gravity
v = u - gt
v² = u² - 2gs
21s ut gt
2
n
2n 1S u g
2
Distance traveled in nth second of fall
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Class Exercise
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Class Exercise -1Which of the following graphs defines the motion of an object thrown vertically up with a velocity V from a height H above the ground?
Tim e
Dis
plac
emen
t
t 2t
(a)
Tim e
Dis
plac
emen
t
t 2t
(b)
Tim e
Dis
plac
emen
t
t1 t2
t2 > 2t1
(c)
Time
Dis
plac
emen
t
t1 t2
t2 > 2t
(d)
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Solution
Tim e
Dis
plac
emen
t
t1 t2
t2 > 2t1
Hence answer is (c)
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Class Exercise -2
A stone is dropped from a rising balloon at a height of 76 m above the ground and reaches the ground in 6 s. What is the approximate velocity of the balloon when the stone was dropped?
52(a) m / s upwards
3
52(b) m / s downwards
3
(c) 3 m/s (d) 9.8 m/s
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Solution
u Initia l position
Final position
76 m
If u is the velocity of balloon in the upward direction when the stone was dropped, then
2176 u(6) 10 (6 )
2
–76 = 6u – 180
52u m/s
3
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Class Exercise - 3A particle moves in a straight line with a velocity v = –3t2 + 2t + 5 ms–1, where t is the time. Assuming x(t) = 0 at t = 0, find
(i) acceleration a(t), at t = 2 s,
(ii) position of particle at t = 5 s.
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Solution
a(2) = –6 × 2 + 2 = –10 ms–2
(ii)dx
vdt
0
t
t
x v t dt
5
2
0
x 5 3t 2t 5 dt
53 2
0t t 5t = –75 m
dva 6t 2
dt (i)
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Class Exercise - 4
What is the time taken by an object thrown vertically upwards at 50 m/s to attain a displacement which is half of the maximum displacement?
(Take g = 10 m/s2)
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Solution
2
maxu 50 50
S 125 m2g 2 10
max
2
s 62.5 m
2162.5 50t – 10 t
2
212.5 10t – t
2t2 – 20t + 25 = 0
40 400 – 200t
4
40 200 20 5 2 20 – 5 2 s or s
4 2 2
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Class Exercise - 5
For an object moving in one dimension, find a if v = 3x2 + 4x m/s.
2v 3x 4x
dv dv dx
dx dt dt
(6x 4) a v
2
26x 4
a m/s3x 4
Solution :
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Class Exercise - 6
a a
u u
v = ?m
A mass m is being pulled up using two strings passing over two pulleys so that the downward velocities of the strings are equal to u. (see figure). What is the speed v of the mass ?
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Solution
v cos = u
uv
cos
a a
v
u uu u
m
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Class Exercise - 7
Two objects are thrown, one upwards and one downwards with a speed u from the top of a tower. If the time of flights are t1 and t2, find the time of flight of an object dropped from the tower.
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SolutionThe time of flight to start from P and come back to that point is t1 – t2 for the ball thrown upwards.
u
u
P
t – t1 2
1 22u
(t – t )g
1 2g
u (t – t )2
22 2
1Height of the tower H ut gt
2
2
1 2 2 22
gt t gt 1– gt
2 2 2 1 2gt t
2
Hence, time of flight for a ball that is dropped
2H
g 1 2t t s
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Class Exercise - 8An object is thrown upwards from the top of a tower with a velocity v1. At the same instance another object is thrown from the bottom of the tower with a velocity v2. If they meet at the top of the tower, find the ratio v1 : v2. (Assume v2 is just enough to reach the top of the tower.)
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Solution
11
2vt
g
22
vt
g
v1 : v2 = 1 : 2
Given that t2 = t1
v2
v1 t1
t2
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Class Exercise - 9
v 10 i 6 j m/s ; u 2 i m/s. Find a if t 3s.
Along X-axis: x10 – 2 8
a3 3
Along Y-axis: y6 – 0
a 23
x ya a i a j
28
i 2 j m/s3
Solution :
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Class Exercise - 10
2v 10 i 6 j m/s ; u 2 i m/s ; a x i +y j m/s
Find x and y if t = 5 s.
Use the same method as question 9.
8 6
x m; y m5 5
Solution :
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Thank you