Problems Linear motion

Problems Linear motion

In this post and  in few of my posts to come, I would like to solve problems on linear motion,freely falling bodies,vertically projected up bodies and projectiles .

1.An object accelerates from rest to a velocity 20m/sec in 4seconds.If  the   object  has uniform acceleration, find its acceleration and displacement in this time.

Soln: From the data given in the problem we have,
Initial velocity = u =0,
final velocity v=20 m/sec,
Time of journey t=4sec,
Acceleration a = \frac{(v-u)}{t} = \frac{(20-0)}{4}= 5 msec^{-2}
Displacement S = ut + \frac{1}{2} at^2 = 0 (4)+\frac{1}{2}\times5\times4^2
S = 40m.

2.An object starting from rest moves with uniform acceleration of 3msec^{-2} for 6sec.Find its velocity and displacement after 6seconds.

Soln: From the data given in the problem we have,
Initial velocity of the object u = 0,
Acceleration of the object a= 3msec^{-2},
Time of journey t =6sec.
Final velocity of the object v=u+at =0 + 3(6) = 18 m/sec.
Displacement S= ut + \frac{1}{2} at^2 = 0 (6)+\frac{1}{2}\times3\times6^2
S= 54m.

3.An object starting from rest moves with uniform acceleration of 4msec^{-2}.Find its displacement i) 5seconds   ii) in 5th second iii) 8th second.

Soln : From data given in the problem
Initial velocity of the object u=0,
Acceleration of the object a=4msec^{-2},
i) time t=5 seconds,
Displacement of the object S=ut + \frac{1}{2} at^2 = 0 (5)+\frac{1}{2}\times4\times5^2
Displacement of the object in 5seconds  S= 50m.
ii) Displacement of the body in 5th second =?
Let us substitute n=5 in the formula S_n=u+a(n-1/2)
S_5=0+4(5-1/2) = 4(4.5) =18m.
iii)Displacement of the body in 8th second =?
Let us substitute n=8 in the formula S_n=u+a(n-1/2)
S_8=0+4(8-1/2) = 4(7.5) =30m.

4.An object started moving with an initial velocity of 10m/sec, after traveling a distance of 5m  gets a velocity 20m/sec.Find its i) acceleration ii) time taken for 5m displacement.

soln: From the data given in the problem,
Initial velocity of the object u=10m/sec,
Final velocity of the object    v= 20m/sec,
Displacement S=5m,
i) acceleration a=?
Substitute the values of u,v and S in the equation V^2-U^2 =2as,
we get (20)^2-(10)^2=2a(5)
300 = 10a  or a = 300/10=30msec^{-2}.
ii)Time t=?
Substitute the values of u,v and a in the equation v=u+at,
we get  20=10+(30)t ;  10=30t
t =10/30 = 1/3 = 0.333 sec.

5.When an observer started observing a car it’s velocity was x m/sec , if it travels for 10sec with uniform acceleration 2.5msec^{-2}and its velocity increases to 75m/sec.Find i) Initial velocity of the car ii) Displacement of the car in 10sec iii) Displacement of the car in first 5sec and last 5sec, what is your inference.
Soln: From the data given in the problem,
Initial velocity of the car u = x (say),
Final velocity = 75m/sec,
Acceleration a = 2.5msec^{-2},
Time of journey t=10sec.
i) Substitute the values of u,v,a and t in the equation v=u+at
we get   75 = x+2.5(10) ; x=50 m/sec.
ii) Let the displacement of the car in 10sec  be S
Substitute the values of u,a and t in the equation S=ut +\frac{1}{2} at^2
We get    S = 50(10)+\frac{1}{2} (2.5)(10)^2 ;
S = 500+125 ; s=675 m
ii) Let the displacement in first 5sec be S_1
Substitute t=5 sec and thve values of u and a in the equation S_1=ut +\frac{1}{2} at^2
we get S_1=50(5) +\frac{1}{2} (2.5)(5)^2
S_1 = 250 + 31.25 = 281.25 m.
Let the displacement in  next  5sec be S_2
S_2 = S –S_1
S_2 = 675 – 281.25 = 393.75 m
We can observe that, even though the time of journey is same , S_2>S_1
Displacement of the body in second half S_2 is greater than in the first half time S_1 of its journey.

6 .A cheetah can accelerate from rest to 24.0 m/s in 6.70 s.
Assuming constant acceleration, how far has the cheetah run in this time?(Question by suzi in yahoo answers).
Soln: From the data given in the problem,
Initial velocity of the cheetah  u=0,
Final velocity of cheetah   v=24 m/sec,
Time t = 6.7sec
Distance traveled by cheetah  be S=?
The equations of motion are v^2-u^2 = 2aS- – – – –   (1)
and  a = \frac{(v-u)}{t} – – –  –  –  (2)
From equations (1) and (2) we get v^2-u^2=2\frac{(v-u)}{t}S
simplifying it we get   S = \frac{(v^2-u^2)}{2}\frac{t}{(v-u)} = \frac{(v+u)}{2}\times t
Substitute the values of u,v and t    we get S = \frac{(24+0)}{2}\times6.7 =12(6.7)=80.4 m

7) A car covers first halt of the distance between two places at a speed of 30 km/hr and the second half at a 90km/hr.What will be the average speed of the car?

Method I: Let the total distance between the places be S.
Time taken to cover First half t_1= \frac{S}{2\times30} = \frac{S}{60} hours.
Time taken to cover Second  half t_1= \frac{S}{2\times90} = \frac{S}{180} hours.
Total time   t = t_1+ t_2 =\frac{S}{60}+\frac{S}{180}=\frac{4S}{180} = \frac{S}{45}.
Average speed = Total distance /total time.
= S/t = \frac{(S)(45)}{S}
= 45km/hr.
Method II (Short cut):  If the body covers 1st half of distance with a speed x and the second half with a speed y,then the average speed = \frac{2xy}{x+y}.
Average speed  = \frac{2(30)(90)}{30+90} = latex \frac{2700}{60}$ = 45km/hr.

8)  A body starts from rest and acqires a velocity of 400m/sec  in 10seconds.Calculate the acceleration and distance traveled.

Soln: From the data given in the problem
Initial velocity of the body u=0,
Time of journey   t=10sec,
Final velocity v=400m/sec,
Acceleration  a=?  and distance traveled in 10sec   s=?
Substitute the values of u,v and t in the equation   v=u+at,
we get     400= (0)(10) + a (10) ;  10a=400
a= 40m/sec^2
Substitute the  values of u,a and t in the equation S= ut +\frac{1}{2} at^2
S=(0)(10)+\frac{1}{2} (40)(10)^2 = 0+2000 = 2000m.

9) A body moving with uniform acceleration covers 6m in 2^{nd} second and 16m in 4^{th} second.Calculate the initial velocity,acceleration and distance moved in 6^{ th } second .
Soln: From the data given in the problem
Distance moved in 2^{nd} second s_2 = 6m,
Distance moved in 4^{th} second s_4 = 16m,
s_4s_2 = a(4-2) =2a
therefore 2a = s_4s_2=16-6 =10
a = 5 m/s^2
Substitute the values of s_2,a  and n=2 in the equation s_2 =u+a(n -1/2)
we get    6=u+5(2-1/2) ; 6=u+7.5
u=-7.5 +6 = -1.5 m/sec.

Distance moved in the 6^{th} second = u+a(6-1/2),

Substitute the values of   u,a  in the above equation we get 6^{th} second=-1.5+5(5.5)
6^{th} second =-1.5+27.5 =26m.

10) An object started traveling with a velocity 2m/sec moves with an uniform acceleration of 3 m/s^2.
i) Find the ratio of displacements in   a) 1^{st},3^{rd},5^{th} seconds  b) 2^{nd},4^{th},  and 6^{th} seconds .
ii)Find the ratio of velocities   a) 1^{st},3^{rd},5^{th} seconds  b) 2^{2d},^{th},6^{th} seconds .

Soln: From the data given in the problem
Initial velocity  u=2 m/sec,
Acceleration   a =3 m/sec^2,
i- a) The ratio of displacements in 1^{st},3^{rd},5^{th} seconds
From the formula s_1:s_3:s_5 = (2u+a) : (2u+5a) : (2u+9a)
s_1:s_3:s_5 = (4+3) : (4+15) : (4+27)
s_1:s_3:s_5 = 7:19:31 .
i-b) The ratio of displacements in 2^{nd},4^{th},6^{th} seconds
From the formula s_2:s_4:s_6 = (2u+3a) : (2u+7a) : (2u+11a)
s_2:s_4:s_6 = (4+9) : (4+21) : (4+33)
s_2:s_4:s_6 = 13 : 25 : 37 .
ii -a) From the formula v_1:v_2:v_3: .  .  .  .  .  .  .  . :v_n= (u+a) : (u+2a) : (u+3a) : .   .   .  .  .  .  .  .  . : (u+na).
The ratio of velocities   a) 1^{st},3^{rd},5^{th} seconds
v_1:v_3:v_5 = (u+a) : (u+3a) : (u+5a) = 5 :11 : 17 .
ii- b) The ratio of velocities   a) 2^{nd},4^{th},6^{th} seconds
v_2:v_4:v_6 = (u+2a) : (u+4a) : (u+6a) =8 :1 4 :20 .

11) A body travels 200cm in the first two seconds and 220cm in the next four seconds.What will be the velocity at the end of the seventh second from the start?

Soln: The displacement of the body in first 2 sec S_1 =200cm,
Let the initial velocity = u(say) ,
Acceleration = a(say), time t_1 =2sec
S_1 = ut_1+\frac{1}{2} at_1^2,
Substitute the value of t_1 in above equation, we get
S_1 = 2u+\frac{1}{2} a(2)^2,
S_1 = 2u +2a ; 2(u+a) =200
Therefore          u+a = 100 – – – – – – – – – – – – –  – – – – –  – (1)
Given that the body travels 220cm in next 4sec.That is from the start it displaces  200+220 = 420 cm in  6sec.
Displacement S_2 =420 cm,
time t_2=6 sec,
substitute theses values in the equation S_2 = ut_2+\frac{1}{2} at_1^2,
S_2 = 6u+\frac{1}{2} a(6)^2=6u+18a,
6(u+3a) = 420 ; u+3a = 70 – – – – – – – – – – – – –  – – – – –  – (2)
Solving equations (1) and (2)     or Eq (2) – Eq(1)
we get  2a= -30 ;     a=-15 cm/sec^2,
Substitute value of  a in Eq(1) we get  u-15 = 100,
u = 115 cm/sec.
The velocity at the end of 6{th} second  v=u+at_2
we get    v= 115+(-15)(6) =115-90 =25cm/sec.
Therefore  final velocity v=25cm/sec.

12.A subway train starts from rest at a station and accelerates at a rate of 16.5m/sec^2 for  13.1 sec.It runs at constant speed for 69.7s and slows down at a rate of 3.45 m/sec^2until it stops at the next station.What is the total distance covered?

Soln: From the data given in the problem,
Initial velocity of the train u = 0 m/sec,
Acceleration a = 16.5 m/sec^2,
time t = 13.1 sec,
Velocity after 13.1 sec v=?
and the distance traveled S_1 =?
Substitute  the values in the equation v =u+at
we get v= 0+(16.5)(13.1) = 216.15 m/sec.
Substitute in equation S_1 = ut + \frac{1}{2} at^2
we get S_1 = (0)(13.1)+\frac{1}{2} (16.5)(13.1)^2
S_1 = 0+1415.78 =1415.78 m
After that the train travels with velocity v=216.15 m/sec for  69.7sec. calculate distance traveled S_2 during this time.
v=216.15 m/sec, t = 69.7 sec S_2=?
substitute the values in equation S_2= vt = (216.15)(69.7)=15065.66 m
Finally the trains decelerates  at the rate of 3.45 m/sec^2 and comes to rest.find distanceS_3 traveled before coming to rest.
Acceleration a = -3.45 m/sec^2 ,
Initial velocity u = 216.15m/sec,
Final velocity v=0,
distance traveled S_3 =?
substitute the values in v^2 - u^2 = 2as,
we get 0^2 - 216.15^2 = 2(-3.45) S_3
-6.90 S_3 = -46720.82
S_3 = 6771.13.
Total distance traveled by train S = S_1 +S_2+ S_3 = 1415.78+15065.66+6771.13=23252.57 m  or 23.252 km.

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