Kinematics-05- Subective UnSolved

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LEVEL - I (C.B.S.E. LEVEL) (REVIEW YOUR CONCEPTS) 1. A particle is moving in the plane with velocity given by ˆ ˆ , where ˆ ˆ are unit U  U0i  a cos t j vectors along x & y axes repectively. If particle is at the origin at t = 0. (a) calculate the trajectory of the particle. i & j (b) find its distance from the origin at time 3 2 . 2. A cricketer hits a ball from the ground level with a velocity →  (20ˆi  10ˆj) m/sec. Find the velocity of the ball at t = 1sec, from the instant of projection (g = 10 m/sec2). 3. A particle starts moving due east with a velocity v1  5 m/sec. For 10 sec. And turns to north with a velocity v 2  10 m/sec. For 5 sec. Find the average velocity of the particle during 15 sec. From start- ing. 4. A football player kicks the football so that it will have a “hang time” (time of flight) of 5s and lands 50 m away. If the ball leaves the player’s foot 1.5m above the ground, what is its initial velocity (magnitude and direction)? (g = 10 m/sec2). 5. Two balls are projected from the same point in directions inclined at 60º and 30º to horizontal. If they attain the same height, what is the ratio of their velocities of projection? What is this ratio of they have the same horizontal range? 6. At what angle should a body be projected with a velocity 24 m/s just to pass over an obstacle 16m high at a horizontal distance of 32 m? (Assume g = 10 m/s2) 7. Two stones are projected from a point P in the same side with a velocity of 20 m/s one at an angle  above the horizontal and the other at angle  to the vertical. Find the distance of each stone from P one second after projection and the distance that stones are apart. (Take cos   4 / 5 . sin   3 / 5 ) 8. A racing car moving with constant acceleration covers two successive kilometeres in 30 s and 20 s respectively. Find the acceleration of the car and initial speed. 9. A particle is projected vertically upwards. Prove that it will be at ¾ of its greatest height at times, which are in the ratio 1 : 3. 10. An aircraft flying horizontally at 360 km/hr releases a bomb (with no velocity relative to the aircraft) at a stationary tank 200 m away. What must be the height of the aircraft above the tank if the bomb is to hit the tank? 11. From the velocity-time plot shown in figure, find the distance trav- eled by the particle during the first 40 s. Also find the average velocity during this period. 5 m/ -5 m/ 12. Two trains each of length 90 m moving in opposite directions along parallel tracks meet when their speeds are 60 km/h and 40 km/hr. If their accelerations are 0.3 m/s2 and 0.15 m/s2 respectively, find the time they take to pass each other. 13. A gun kept on a straight horizontal road is used to hit a car travelling along the same road away from the gun with a uniform speed of 72 km/hr. The car is at a distance of 500 m from the gun which the gun is fired at an angle of 45º with the horizontal. Find (i) the distance

LEVEL - I (C.B.S.E. LEVEL)
(REVIEW YOUR CONCEPTS)
1. A particle is moving in the plane with velocity given by j
ˆ
t
cos
a
î
U
U 0 




, where j
ˆ
&
î are unit
vectors along x & y axes repectively. If particle is at the origin at t = 0.
(a) calculate the trajectory of the particle.
(b) find its distance from the origin at time


2
3
.
2. A cricketer hits a ball from the ground level with a velocity )
ĵ
10
î
20
(
v0 


m/sec. Find the velocity of
the ball at t = 1sec, from the instant of projection (g = 10 m/sec2
).
3. A particle starts moving due east with a velocity 5
v1  m/sec. For 10 sec. And turns to north with a
velocity 10
v2  m/sec. For 5 sec. Find the average velocity of the particle during 15 sec. From start-
ing.
4. A football player kicks the football so that it will have a “hang time” (time of flight) of 5s and lands 50
m away. If the ball leaves the player’s foot 1.5m above the ground, what is its initial velocity (magnitude
and direction)? (g = 10 m/sec2
).
5. Two balls are projected from the same point in directions inclined at 60º and 30º to horizontal. If they
attain the same height, what is the ratio of their velocities of projection? What is this ratio of they have
the same horizontal range?
6. At what angle should a body be projected with a velocity 24 m/s just to pass over an obstacle 16m high
at a horizontal distance of 32 m? (Assume g = 10 m/s2
)
7. Two stones are projected from a point P in the same side with a velocity of 20 m/s one at an angle 
above the horizontal and the other at angle  to the vertical. Find the distance of each stone from P one
second after projection and the distance that stones are apart. (Take 5
/
4
cos 
 . 5
/
3
sin 
 )
8. A racing car moving with constant acceleration covers two successive kilometeres in 30 s and 20 s
respectively. Find the acceleration of the car and initial speed.
9. A particle is projected vertically upwards. Prove that it will be at ¾ of its greatest height at times, which
are in the ratio 1 : 3.
10. An aircraft flying horizontally at 360 km/hr releases a bomb (with no velocity relative to the aircraft) at
a stationary tank 200 m away. What must be the height of the aircraft above the tank if the bomb is to
hit the tank?
V
-5 m/s
5 m/s
0
20 40 -t(s)
11. From the velocity-time plot shown in figure, find the distance trav-
eled by the particle during the first 40 s. Also find the average
velocity during this period.
12. Two trains each of length 90 m moving in opposite directions along parallel tracks meet when their
speeds are 60 km/h and 40 km/hr. If their accelerations are 0.3 m/s2
and 0.15 m/s2
respectively, find the
time they take to pass each other.
13. A gun kept on a straight horizontal road is used to hit a car travelling along the same road away from the
gun with a uniform speed of 72 km/hr. The car is at a distance of 500 m from the gun which the gun is
fired at an angle of 45º with the horizontal. Find (i) the distance of the car from the gun where the shell
hits it and (ii) the speed of projection of the shell from the gun.
14. A particle is projected with a velocity ag
2 so that it just clears two walls of equal height ‘a’. which
are at a distance 2a apart. Show that the time of passing between the walls is g
/
a
2 .
15. At the instant the traffic light turns green, an automobile starts with a constant acceleration of 2.2 m/s2
.
At the same instant a truck, travelling with a constant speed of 9.5 m/s, overtakes and passes the
automobile.
(a) How far beyond the starting point will the automobile overtake the truck?
(b) How fast will the car be traveling at the instant?
(It is instructive to plot a qualitative graph of ‘x’ versus t for each vehicle.)
LEVEL II
(BRUSH UP YOUR CONCEPTS)
1. 2 bodies move towards each other in a straignt line at initial velocities v1
& v2
and with constant
accelerations a1
& a2
directed against the corresponding velocities at the initial instant. What must be
the maximum initial separation lmax
between the bodies for which they meet during the motion?
2. A projectile is launched at an angle  from a cliff of height H above the sea level. If it falls into the sea
at a distance D from the base of the cliff, show that its maximum height above the sea level is










)
tan
D
H
(
4
tan
D
H
2
2
.
3. A stone is projected from the point of a ground in such a direction so as to hit a bird on the top of a
telegraph post of height h and then attain the maximum height 2h above the ground. If at the instant of
projection, the bird were to fly away horizontally with a uniform speed, find the ratio between the
horizontal velocities of the bird and the stone, if the stone still hits the bird while descending.
4. Find the ratio between the normal and tangential acceleration of a point on the rim of a rotating wheel
when at the moment the vector of the total acceleration of this point forms an angle of 30º with the
vector of the linear velocity.
5. Two ships are resting on sea at distance a & b from a fixed point O respectively. They start moving
towards the point O with constant velocities 1
v and 2
v respectively. If the ships subtend an angle  at
O, find the shortest distance of their separation. (Find the result at )
2
/


 .
6. Two particles moves in a uniform gravitational field with an acceleration g. At the initial moment the
particles were located at one point in space and moved with velocities 0
.
3
v1  m/s and 2
v 4.0m/s

horizontally in opposite directions. Find the separation between the particles at the moment when their
velocity vectors become mutually perpendicular.
7. A projectile crosses half its maximum height at a certain instant of time and again 10 s later. Calculate
the maximum height. If the angle of projection was 30º, calculate the maximum range of the projectile
as well as the horizontal distance it travelled in the above 10s.
8. Two trains having a speed of 30 km/hr are headed at each other on the same straight track. A bird that
can fly at 60 km/hr files off one train when they are 60 km apart and heads directly for the other train.
On reaching the other train it files directly back to the first and sot forth. Find
(a) How many trips can the bird make from one train to the other before the trains collide?
(b) What is the total distance travelled by the bird?
9. A projectile is projected with a velocity u at an angle  to the horizontal, in the vertical plane. If after
timet, it is moving in a direction making an angle  with thehorizontal, provethat )
sin(
u
cos
gt 



 .
10. A particle is projected with a velocity u at an angle  with the horizontal. Find the radius of the
curvature of the parabola traced out by the particle at the point where velocity makes an angle
2

with
the horizontal.
LEVEL III
(CHECK YOUR SKILLS)
1. Two particles are simultaneously thrown from roofs of two high build-
ings, as shown in fig. Their velocities of projection are 2 ms-1
and 14
ms-1
respectively. Horizontal and vertical separation between points
A and B is 22 m and 9 m respectively. Calculate minimum separation
between the particles in the process of their motion.
2. Two ships A and B originally at a distance d from each other depart at the same time from a straight
coastline. Ship A moves along a straight line perpendicular to the shore while ship B constaantly heads
for ship A, having at each moment the same speed as the latter. After a sufficiently great interval of
time the second ship will obviously follow that first one at a certain distance. Find the distance.
3. A cannon fires successively two shells with velocity 250
v0  m/s; the first at the angle º
60
1 
 and
the second at the angle º
45
2 
 to the horizontal, the azimuth being the same. Neglecting the air drag,
find the time interval between firings leading to the collision of the shells.
y
1.25m
u
O 3.0 m
A
P
4. An object A is kept fixed at the point x = 3m and y = 1.25 m
on a plank P raised above the ground. At time t = 0 the plank
starts moving along the +x direction with an acceleration 1.5
m/s2
. At the same instant a stone is projected from the origin
with a velocity u as shown.Astationary person on the ground
observes the stone hitting the object during its downward
motion at an angle of 45º to the horizontal.All the motions are
in the x – y plane. Find u and the time after which the stone
hits the object. Take g = 10 m/s2
.
5. A bottle was released from rest from a height of 60 m above the ground. Simultaneously, a stone was
thrown from a point on the ground 60 m distant horizontally from the bottle, with a velocity u at anangle
of projection of  , in a vertical plane containing the bottle. If the stone strikes the bottle 3 s after the
instant of projection, find the velocity u and the angle  of projection.
6. An aeroplane flies horizontally at height h at a constant speed v. An anti-aircraft gun fires a shell at the
22m
A 45º
u = 2 ms-1
v = 14 ms-1
9m
B
45º
plane when it is vertically above the gun. Show that the minimum muzzle velocity of the shell required
to hit the plane is gh
2
v2
 at an angle 1
tan ( 2gh / v)

.
7. Two particles start simultaneously from the same point and move along two straight lines, one with
uniform velocity u and the other with constant acceleration f. Show that their relative velocity is least
after time 




 
f
cos
u
and the least relative velocity is 
sin
u . Where  is the angle between the lines.
8. Two boats Aand B move away from buoy anchored at the middle of a river along mutually perpendicu-
lar straight lines, the boat A along the river and the boat B across the river. Having moved off an equal
distance from the buoy the boat returned. Find the times of motion of boats B
A t
/
t if the velocity of
each boat with respect to water is n times greater than the stream velocity.
u
B
A
P
Q
h
30º 60º
O
9. Two inclined planes OAand OB having inclination with horizon-
tal) 30º and 60º respectively, intersect each other at O as shown
in figure. A particle is projected from point P with velocity
3
10
u  m/s Along a direction perpendicular to plane OA. If
the particle strikes plane OB perpendicularly at Q, calculate
(a) Velocity with which particle strikes the plane OB,
(b) Timeof flight,
(c) Vertical height h of P from O,
(d) Maximum height from O, attained by the particle and
(e) distance PQ, )
s
/
m
10
g
( 2

10. A balloon starts rising from the surface of the Earth. The ascent rate is constant and equal to 0
v . Due
to the wind the balloon gathers the horizontal velocity component x
v = ay, where a is a constant and y
is the height of ascent. Find how the following quantities depend on the height of ascent.
(a) The horizontal drift of the balloon x(y);
(b) The total, tangential, and normal accelerations of the balloon.

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Kinematics-05- Subective UnSolved

  • 1. LEVEL - I (C.B.S.E. LEVEL) (REVIEW YOUR CONCEPTS) 1. A particle is moving in the plane with velocity given by j ˆ t cos a î U U 0      , where j ˆ & î are unit vectors along x & y axes repectively. If particle is at the origin at t = 0. (a) calculate the trajectory of the particle. (b) find its distance from the origin at time   2 3 . 2. A cricketer hits a ball from the ground level with a velocity ) ĵ 10 î 20 ( v0    m/sec. Find the velocity of the ball at t = 1sec, from the instant of projection (g = 10 m/sec2 ). 3. A particle starts moving due east with a velocity 5 v1  m/sec. For 10 sec. And turns to north with a velocity 10 v2  m/sec. For 5 sec. Find the average velocity of the particle during 15 sec. From start- ing. 4. A football player kicks the football so that it will have a “hang time” (time of flight) of 5s and lands 50 m away. If the ball leaves the player’s foot 1.5m above the ground, what is its initial velocity (magnitude and direction)? (g = 10 m/sec2 ). 5. Two balls are projected from the same point in directions inclined at 60º and 30º to horizontal. If they attain the same height, what is the ratio of their velocities of projection? What is this ratio of they have the same horizontal range? 6. At what angle should a body be projected with a velocity 24 m/s just to pass over an obstacle 16m high at a horizontal distance of 32 m? (Assume g = 10 m/s2 ) 7. Two stones are projected from a point P in the same side with a velocity of 20 m/s one at an angle  above the horizontal and the other at angle  to the vertical. Find the distance of each stone from P one second after projection and the distance that stones are apart. (Take 5 / 4 cos   . 5 / 3 sin   ) 8. A racing car moving with constant acceleration covers two successive kilometeres in 30 s and 20 s respectively. Find the acceleration of the car and initial speed. 9. A particle is projected vertically upwards. Prove that it will be at ¾ of its greatest height at times, which are in the ratio 1 : 3. 10. An aircraft flying horizontally at 360 km/hr releases a bomb (with no velocity relative to the aircraft) at a stationary tank 200 m away. What must be the height of the aircraft above the tank if the bomb is to hit the tank? V -5 m/s 5 m/s 0 20 40 -t(s) 11. From the velocity-time plot shown in figure, find the distance trav- eled by the particle during the first 40 s. Also find the average velocity during this period. 12. Two trains each of length 90 m moving in opposite directions along parallel tracks meet when their speeds are 60 km/h and 40 km/hr. If their accelerations are 0.3 m/s2 and 0.15 m/s2 respectively, find the time they take to pass each other.
  • 2. 13. A gun kept on a straight horizontal road is used to hit a car travelling along the same road away from the gun with a uniform speed of 72 km/hr. The car is at a distance of 500 m from the gun which the gun is fired at an angle of 45º with the horizontal. Find (i) the distance of the car from the gun where the shell hits it and (ii) the speed of projection of the shell from the gun. 14. A particle is projected with a velocity ag 2 so that it just clears two walls of equal height ‘a’. which are at a distance 2a apart. Show that the time of passing between the walls is g / a 2 . 15. At the instant the traffic light turns green, an automobile starts with a constant acceleration of 2.2 m/s2 . At the same instant a truck, travelling with a constant speed of 9.5 m/s, overtakes and passes the automobile. (a) How far beyond the starting point will the automobile overtake the truck? (b) How fast will the car be traveling at the instant? (It is instructive to plot a qualitative graph of ‘x’ versus t for each vehicle.) LEVEL II (BRUSH UP YOUR CONCEPTS) 1. 2 bodies move towards each other in a straignt line at initial velocities v1 & v2 and with constant accelerations a1 & a2 directed against the corresponding velocities at the initial instant. What must be the maximum initial separation lmax between the bodies for which they meet during the motion? 2. A projectile is launched at an angle  from a cliff of height H above the sea level. If it falls into the sea at a distance D from the base of the cliff, show that its maximum height above the sea level is           ) tan D H ( 4 tan D H 2 2 . 3. A stone is projected from the point of a ground in such a direction so as to hit a bird on the top of a telegraph post of height h and then attain the maximum height 2h above the ground. If at the instant of projection, the bird were to fly away horizontally with a uniform speed, find the ratio between the horizontal velocities of the bird and the stone, if the stone still hits the bird while descending. 4. Find the ratio between the normal and tangential acceleration of a point on the rim of a rotating wheel when at the moment the vector of the total acceleration of this point forms an angle of 30º with the vector of the linear velocity. 5. Two ships are resting on sea at distance a & b from a fixed point O respectively. They start moving towards the point O with constant velocities 1 v and 2 v respectively. If the ships subtend an angle  at O, find the shortest distance of their separation. (Find the result at ) 2 /    . 6. Two particles moves in a uniform gravitational field with an acceleration g. At the initial moment the particles were located at one point in space and moved with velocities 0 . 3 v1  m/s and 2 v 4.0m/s  horizontally in opposite directions. Find the separation between the particles at the moment when their velocity vectors become mutually perpendicular. 7. A projectile crosses half its maximum height at a certain instant of time and again 10 s later. Calculate the maximum height. If the angle of projection was 30º, calculate the maximum range of the projectile as well as the horizontal distance it travelled in the above 10s.
  • 3. 8. Two trains having a speed of 30 km/hr are headed at each other on the same straight track. A bird that can fly at 60 km/hr files off one train when they are 60 km apart and heads directly for the other train. On reaching the other train it files directly back to the first and sot forth. Find (a) How many trips can the bird make from one train to the other before the trains collide? (b) What is the total distance travelled by the bird? 9. A projectile is projected with a velocity u at an angle  to the horizontal, in the vertical plane. If after timet, it is moving in a direction making an angle  with thehorizontal, provethat ) sin( u cos gt      . 10. A particle is projected with a velocity u at an angle  with the horizontal. Find the radius of the curvature of the parabola traced out by the particle at the point where velocity makes an angle 2  with the horizontal. LEVEL III (CHECK YOUR SKILLS) 1. Two particles are simultaneously thrown from roofs of two high build- ings, as shown in fig. Their velocities of projection are 2 ms-1 and 14 ms-1 respectively. Horizontal and vertical separation between points A and B is 22 m and 9 m respectively. Calculate minimum separation between the particles in the process of their motion. 2. Two ships A and B originally at a distance d from each other depart at the same time from a straight coastline. Ship A moves along a straight line perpendicular to the shore while ship B constaantly heads for ship A, having at each moment the same speed as the latter. After a sufficiently great interval of time the second ship will obviously follow that first one at a certain distance. Find the distance. 3. A cannon fires successively two shells with velocity 250 v0  m/s; the first at the angle º 60 1   and the second at the angle º 45 2   to the horizontal, the azimuth being the same. Neglecting the air drag, find the time interval between firings leading to the collision of the shells. y 1.25m u O 3.0 m A P 4. An object A is kept fixed at the point x = 3m and y = 1.25 m on a plank P raised above the ground. At time t = 0 the plank starts moving along the +x direction with an acceleration 1.5 m/s2 . At the same instant a stone is projected from the origin with a velocity u as shown.Astationary person on the ground observes the stone hitting the object during its downward motion at an angle of 45º to the horizontal.All the motions are in the x – y plane. Find u and the time after which the stone hits the object. Take g = 10 m/s2 . 5. A bottle was released from rest from a height of 60 m above the ground. Simultaneously, a stone was thrown from a point on the ground 60 m distant horizontally from the bottle, with a velocity u at anangle of projection of  , in a vertical plane containing the bottle. If the stone strikes the bottle 3 s after the instant of projection, find the velocity u and the angle  of projection. 6. An aeroplane flies horizontally at height h at a constant speed v. An anti-aircraft gun fires a shell at the 22m A 45º u = 2 ms-1 v = 14 ms-1 9m B 45º
  • 4. plane when it is vertically above the gun. Show that the minimum muzzle velocity of the shell required to hit the plane is gh 2 v2  at an angle 1 tan ( 2gh / v)  . 7. Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity u and the other with constant acceleration f. Show that their relative velocity is least after time        f cos u and the least relative velocity is  sin u . Where  is the angle between the lines. 8. Two boats Aand B move away from buoy anchored at the middle of a river along mutually perpendicu- lar straight lines, the boat A along the river and the boat B across the river. Having moved off an equal distance from the buoy the boat returned. Find the times of motion of boats B A t / t if the velocity of each boat with respect to water is n times greater than the stream velocity. u B A P Q h 30º 60º O 9. Two inclined planes OAand OB having inclination with horizon- tal) 30º and 60º respectively, intersect each other at O as shown in figure. A particle is projected from point P with velocity 3 10 u  m/s Along a direction perpendicular to plane OA. If the particle strikes plane OB perpendicularly at Q, calculate (a) Velocity with which particle strikes the plane OB, (b) Timeof flight, (c) Vertical height h of P from O, (d) Maximum height from O, attained by the particle and (e) distance PQ, ) s / m 10 g ( 2  10. A balloon starts rising from the surface of the Earth. The ascent rate is constant and equal to 0 v . Due to the wind the balloon gathers the horizontal velocity component x v = ay, where a is a constant and y is the height of ascent. Find how the following quantities depend on the height of ascent. (a) The horizontal drift of the balloon x(y); (b) The total, tangential, and normal accelerations of the balloon.