Chapter 3 Motion in a Straight Line Solutions
Question - 1 : - In which of the following examples of motion, can the body be considered
approximately a point object:
(a) a railway carriage moving without jerks between two stations.
(b) a monkey sitting on top of a man cycling smoothly on a circular track.
(c) a spinning cricket ball that turns sharply on hitting the ground.
(d) a tumbling beaker that has slipped off the edge of a table.
Answer - 1 : -
( a ), ( b ) The size ofthe train carriage and the cap is very small as compared to the distancethey’ve travelled, i.e. the distance between the two stations and the length ofthe race track, respectively. Therefore, the cap and the carriage can beconsidered as point objects.
The size of thebasketball is comparable to the distance through which it bounces off afterhitting the floor. Thus, basketball cannot be treated as a point object.Likewise, the size of the bottle is comparable to the height of the chair fromwhich it drops. Thus, the bottle cannot be treated as a point object.
Question - 2 : - The position-time (x-t) graphs for two children A and B returning fromtheir school
O to their homes P and Q respectivelyare shown in Fig. 3.19. Choose the correct
entries in the brackets below ;
(a) (A/B) lives closer to the schoolthan (B/A)
(b) (A/B) starts from the schoolearlier than (B/A)
(c) (A/B) walks faster than (B/A)
(d) A and B reach home at the(same/different) time
(e) (A/B) overtakes (B/A) on the road(once/twice).
Answer - 2 : -
(a) A lives closer tothe school than B, because A has to cover shorter distances [OP < OQ],
(b) A starts from school earlier than B, because for x= 0, t = 0 for A but forB, t has some finite time.
(c) The slope of B is greater than that of A, therefore B walks faster than A.
(d) Both A and B will reach their home at the same time.
(e) At the point of intersection, B overtakes A on the roads once.
Question - 3 : - A woman starts from her home at 9.00 am, walks at a speed of 5 km/h on a
straight road up to her office 2.5 kmaway, stays at the office up to 5.00 pm, and
returns home by an auto with a speed of25 km/h. Choose suitable scales and
plot the x-t graph of her motion.
Answer - 3 : -
Distance till heroffice = 2.5 km.
Walking speed the woman= 5 km/h
Time taken to reach office while walking = (2.5/5 ) h=(1/2) h = 30 minutes
Speed of auto = 25km/h
Time taken to reachhome in auto = 2.5/25 = (1/10) h = 0.1 h = 6 minutes
In the graph, O istaken as the origin of the distance and the time, then at t = 9.00 am, x= 0
and at t = 9.30 am, x = 2.5 km
OA is the portion onthe x-t graph that represents her walk from home to the office. AB representsher time of stay in the office from 9.30 to 5. Her return journey isrepresented by BC.
Question - 4 : - A drunkard walking in a narrow lane takes 5 steps forward and 3 stepsbackwards, followed again by 5 steps forward and 3 steps backwards, and soon. Each step is 1m long and requires 1 s. Plot the x-t graph of his motion.Determine graphically and otherwise how long the drunkard takes to fall ina pit 13 m away from the start.
Answer - 4 : -
The time taken to goone step is 1 second. In 5s he moves forward through a distance of 5m and thenin next 3s he comes back by 3m. Therefore, in 8s he covers 2m. So, tocover a distance of 8m he takes 32s. He must take another 5steps forward tofall into the pit. So, the total time taken is 32s + 5s = 37s to fall into apit 13 m away.
Question - 5 : - A jet airplane travelling at the speed of 500 km/h ejects its products ofcombustion at the speed of 1500 km/h relative to the jet plane. What isthe speed of the latter with respect to an observer on the ground?
Answer - 5 : -
Speed of the jetairplane, VA= 500 km/h
Speed at which thecombustion products are ejected relative to the jet plane, VB –VA
= – 1500 km/h
(The negative sign indicatesthat the combustion products move in a direction opposite to that of jet)
Speed of combustionproducts w.r.t. observer on the ground, VB – 500 = – 1500
VB = –1500 + 500 = – 1000 km/h
Question - 6 : - A car moving along a straight highway with aspeed of 126 km h–1 is brought to a stop within a distanceof 200 m. What is the retardation of the car (assumed uniform), and howlong does it take for the car to stop?
Answer - 6 : -
The initial velocityof the car = u
Final velocity of thecar = v
Distance covered bythe car before coming to rest = 200 m
Using the equation,
v = u + at
t = (v – u)/a = 11.44sec.
Therefore, it takes11.44 sec for the car to stop.
Question - 7 : - Two trains A and B of length 400 m each are moving on two parallel trackswith a
uniform speed of 72 km h–1 inthe same direction, with A ahead of B. The driver of
B decides to overtake A and acceleratesby 1 m s–2. If after 50 s, the guard of B just
brushes past the driver of A, what wasthe original distance between them?
Answer - 7 : -
Length of the train Aand B = 400 m
Speed of both thetrains = 72 km/h = 72 x (5/18) = 20m/s
Using the relation, s= ut + (1/2)at2
Distance covered bythe train B
SB = uBt+ (1/2)at2
Acceleration, a = 1m/s
Time = 50 s
SB =(20 x 50) + (1/2) x 1 x (50)2
= 2250 m
Ditance covered by thetrain A
SA = uAt+ (1/2)at2
Acceleration, a = 0
SA = uAt = 20 x 50 = 1000 m
Therefore, theoriginal distance between the two trains = SB – SA =2250 – 1000 = 1250 m
Question - 8 : - On a two-lane road, car A is travelling at a speed of 36 km/h. Two cars Band
C approach car A in opposite directionswith a speed of 54 km/h each. At a
certain instant, when the distance ABis equal to AC, both being 1 km, B decides
to overtake A before C does. Whatminimum acceleration of car B is required to
avoid an accident?
Answer - 8 : -
The speed of car A =36 km/h = 36 x (5/8) = 10 m/s
Speed of car B = 54km/h = 54 x (5/18) = 15 m/s
Speed of car C =– 54 km/h = -54 x (5/18) = -15 m/s (negative sign shows B and C are inopposite direction)
Relative speed of Aw.r.t C, VAC= VA – VB = 10 – (-15) =25 m/s
Relative speed of Bw.r.t A, VBA = VB – VA = 15 –10 = 5 m/s
Distance between AB =Distance between AC = 1 km = 1000 m
Time taken by the carC to cover the distance AC, t = 1000/VAC = 1000/ 25 = 40 s
If a is the acceleration,then
s = ut + (1/2) at2
1000 = (5 x 40) +(1/2) a (40) 2
a = (1000 – 200)/ 800= 1 m/s2
Thus, the minimumacceleration of car B required to avoid an accident is 1 m/s2
Question - 9 : - Two towns A and B are connected by regular busservice with a bus leaving in
either direction every T minutes. A mancycling with a speed of 20 km h–1 in the
direction A to B notices that a busgoes past him every 18 min in the direction of
his motion, and every 6 min in theopposite direction. What is the period T of the
bus service and with what speed(assumed constant) do the buses ply on the
road?
Answer - 9 : -
Speed of each bus= Vb
Speed of the cyclist =Vc = 20 km/h
The relative velocityof the buses plying in the direction of motion of cyclist is Vb –Vc
The buses playing in the direction of motion of the cyclist go past him afterevery
18 minute i.e.(18/60) s.
Distance covered = (Vb –Vc ) x 18/60
Since the buses areleaving every T minutes. Therefore, the distance is equal to Vb x(T/60)
(Vb –Vc ) x 18/60 = Vb x (T/60) ——(1)
The relative velocityof the buses plying in the direction opposite to the motion of cyclist is Vb +Vc
The buses go past the cyclist every 6 minutes
18 minute i.e.(6/60) s.
Distance covered = (Vb +Vc ) x 6/60
Therefore, (Vb +Vc )x 6/60 = Vb x (T/60)——(2)
Dividing (2) by (1)
[(Vb –Vc ) x 18/60]/ [(Vb + Vc ) x6/60 ]= [Vb x (T/60)] /[Vb x (T/60)]
(Vb –Vc ) 18/(Vb +Vc ) 6 = 1
(Vb –Vc )3 = (Vb +Vc )
Substituting thevalue of Vc
(Vb –20 )3= (Vb + 20 )
3Vb –60 = Vb + 20
2Vb =80
Vb =80/2 = 40 km/h
To find the value ofT, substitute the values of Vb and Vc inequation (1)
(Vb –Vc ) x 18/60 = Vb x (T/60)
(40 – 20) x (18/60) =40 x (T/60)
T = (20 x 18) /40 = 9minutes
Question - 10 : - A player throws a ball upwards with an initial speed of 29.4 m/s.
(a) What is the direction ofacceleration during the upward motion of the ball?
(b) What are the velocity and accelerationof the ball at the highest point of its
motion?
(c) Choose the x = 0 m and t = 0 s tobe the location and time of the ball at its
highest point, vertically downwarddirection to be the positive direction of
the x-axis, and give the signs of position,velocity and acceleration of the ball
during its upward, and downward motion.
(d) To what height does the ball riseand after how long does the ball return to the
player’s hands? (Take g = 9.8 m s–2 andneglect air resistance).
Answer - 10 : -
(a) The accelerationdue to gravity always acts downwards towards the centre of the Earth.
(b) At the highest point of its motion the velocity of the ball will be zerobut the acceleration due to gravity will be 9.8 m s–2 actingvertically downward.
(c) If we consider the highest point of ball motion as x = 0, t = 0 andvertically downward direction to be +ve direction of the x-axis, then
(i) During upward motion of the ball before reaching the highest point position,x = +ve, velocity, v = -ve and acceleration, a = +ve.
(ii) During the downward motion of the ball after reaching the highest pointposition, velocity and acceleration all the three quantities are positive.
(d) Initial speed of the ball, u= -29.4 m/s
Final velocity of theball, v = 0
Acceleration = 9.8 m/s2
Applying in theequation v2 – u2 = 2gs
0 – (-29.4)2 =2 (9.8) s
s = – 864.36/19.6 = –44.1
Height to which theball rise = – 44.1 m (negative sign represents upward direction)
Considering theequation of motion
v = u + at
0 = (-29.4) + 9.8t
t = 29.4/9.8 = 3seconds
Therefore, the totaltime taken for the ball to return to the player’s hands is 3 +3 = 6s