Kinema Tics

BANSAL CLASSES TARGETIIT JEE 2007 XI (P, Q, R, S) ill

s

C O N T E N T S

KEY CONCEPTS EXERCISE-I EXERCISE -II EXERCISE-III ANSWER KEY

KEY

CONCEPTS

THINGS TO REMEMBER ds ; a = — dv = v — dv v= — — dt dt ds ; s = Jvdt; = Ja dt ; = Ja ds where the symbols have their usual meaning . The equations ofmotion for a body moving in straight line with uniform acceleration, are , 1 * = ut h——=•v t — — r - \(m) 2v - 2_L1 u + 2as (i) v = u + at (ii) s=|—r—11 ' v + u^ (iv) s = u + ^ a ( 2 n - l ) (v) v

2.

u + V

a t

t

a t

2

n

If a body is thrown vertically up with a velocity u in the uniform gravitational field then (neglecting air resistance): (i) Maximum height attained H= (ii) Time of ascent = time of descent = — g 2u (iii) Total time of flight (iv) Velocity of fall at the point of projection=u downwards :

KINEMATIC GRAPH: Slope ofthe displacement time graph at any particular time gives the magnitude ofthe instantaneous velocity at that particular time. Slope ofthe v -1 graph will give the magnitude of the instantaneous acceleration. The area between the v - t graph, the time axis and the ordinates erected at the beginning & end oftime interval considered will represent the total displacement of the body. 5. RELATIVE VELOCITY: (a) Velocity of 'A' relative to 'B' is given by V = V - V • V refers to the velocity which 'A' appears to have as seen by B. The above idea of 1 dimensional relative motion can be extended to motion in 2 dimensions. (b) Angular velocity of A relative to B i.e. co is given by velocityof ArelativetoBinadirectionperpendiculartoAB ®AB AB 6. LEVEL GROUND PROJECTILE MOTION: When abody is thrown obliquely (in a vertical plane) into the uniform gravitationalfieldthen the trajectory (actual path of motion) is a parabola. The horizontal component of velocity ucos a remains unchanged where as vertical component decreases up to the maximum height and then increases. (a) Time taken to reach the height point t ^ usina i y a (minimum (b) Maximum height H - u sm velocity) 2g AB

A

B

AB

AB

=

2

(c) Total time of flight

•

2

=2t

u cos a

H

(d) Horizontal range = (ucos a). T= - (ucos a) (usina)

u sin 2 a 2

u c o s

a

[Figure 1]

>x

(e)' Rm if a = 45° Note that for a given velocity ofprojection & a given horizontal range there are in general two directions of proj ection which are complement ofeach other and are equally inclined to the direction ofthe maximum range. v

^Bansal Classes

Kinematics

[2]

(F) VELOCITY & DIRECTION O F MOTION A T A GIVEN TIME :

VcosB =ucosa Squaring & adding these 2 equations we will get the velocity of the VsinB =usina-gt projectile. Dividing the velocities in y and x directions gives the direction of motion.

( g ) VELOCITY & DIRECTION O F MOTION A T A GIVEN HEIGHT H :

V cos 0 =u cos a on adding V = u - 2 gh V sin 0 =u sin a-2gh_ 2

2

2

2

2

2

2

2

2

2

( h ) EQUATIONS O F MOTION IN VECTOR NOTATION :

(i) V=u+ gt (ii) S=ut+—gt (iii) V = -=u+-gt (V = average velocity vector) 2

2

(i)

av

t

av

2.

EQUATION O F TRAJECTORY :

gx - x tan a 2u cos a v Ry dy Note that — dx represent the direction of motion 7. PROJECTILE UP AN INCLINED PLANE : (a) Total time of flight onthe inclined plane —2/ (a-P) T _ 2ug sincosp // \ s Oblique Proj ection (refer fig-1) y = x tan a -

2

2

/ —

(b)

kf

Range PQ on the inclined plane 2u cosa . sin(a-P) g cos p

PQ

2

gcos (3

2

[sin (2 a - P) - sinP]

N

(d)

W

71 u p ForMaxmimumrange 2 a - P = — =>a= —+— ^ T" Z* Hence the direction for maximum range bisects the angle between the vertical and the inclined plane. R = u

(e)

Greatest distance ofthe projectile from the inclined plane;

(c)

2

g(l+sinP)

max

u sin (a-p)

S = 2g cosp when the projectile is at H, its velocity perpendicular to the plane is zero. 8. PROJECTILE DOWN AN INCLINED PLANE: (a) Time offlight= ' ( P) 2

2

2 u s

(b) (c) (d)

faBansal Classes

n

a +

gcosp Range OP 2u sin(a + p). cosa g cos p u Maximum range= g(l-sinp) 2

2

7C _ p Angle ofproj ection a for maximum range= 4 2

Kinematics

[3]

Q.l

EXERCISE - /

A butterfly is flying with velocity 10 i +12 j m/s and wind is blowing along x axis with velocity u. If butterfly starts motionfromA and after some time reaches point B,findthe value of u. y/x =3/4 12/(10+u)=3/4

y

B

37°

Q. 2

Find the change in velocity of the tip of the minute hand (radius =10 cm) of a clock in 45 minutes.

Q.3

A,B&Care threeobjects each movingwith constant velocity. A's speed is lOm/sec in a direction pQ. The velocity of B relative to A is 6 m/sec at an angle of, cos (15/24) to PQ. The velocity of C relative to B is 12 m/sec in a direction Qp, thenfindthe magnitude of the velocity of C. Rain is falling vertically with a speed of 20 ms" relative to air. A person is running in the rain with a velocity of 5 ms and a wind is also blowing with a speed of 15 ms (both towards east). Find the angle with the vertical at which the person should hold his umbrella so that he may not get drenched.

change in velocity = v2-v1 (vectors) so it is different from average velocity which is x2-x1/t

root 2 *v v=omega*r

-1

Q.4 Q.5

1

-1

-1

The velocity-time graph ofthe particle moving along a straight line is shown. The rate of acceleration and deceleration is constant and it is equal to 5 ms"" . If the s average velocity during the motion is 20 ms , thenfindthe value of t.

A=10 i B-A=3.75 i + 4.68 j C-B=-12 i sum all eq

r-a=20 j a=15 i v=5 i r-v=10 i+ 20j

2

-1

25 sec

vt+v(25-2t)=20*25 v=5t 5t(25-t)=500

Q.6

The fig. shows the v-t graph of a particle moving in straight line. Find the time when particle returns to the starting point.

Q.7

A particle is proj ected in the X-Y plane. 2 sec after proj ection the velocity of the particle makes an angle 45° with the X - axis. 4 sec after projection, it moves horizontally. Find the velocity of projection (use g = 10 ms ).

v

-2

Q.8

A small ball rolls off the top landing of a staircase. It strikes the mid point of the first step and then in y direction mid point of the second step. The steps are smooth & identical in height & width. Find the coefficient of restitution between the ball & the first step.

Q.9

A stone is dropped from a height h. Simultaneously another stone is thrown up from the ground with such a velocity that it can reach a height of 4h. Find the time when two stones cross each other.

t1=root(2h/g) t2=t1*2 v1=root (2gh) -h=v2*t1*2- 1/2*g* (2*t1)^2 v2=root (9gh/8)

relative acceleration=0 u=root (8gh)

Q.10 A particle is proj ected upwards with a velocity of 100 m/sec at an angle of 60° with the vertical. Find the time when the particle will move perpendicular to its initial direction, taking g=10 m/sec . 2

Q.ll A particle is moving on a straight line. Its displacementfromthe initial position |s„j is plotted against time in the graph shown. What will be the velocity of the particle at 2/3 sec? Assume the graph to be a sine curve. - / x=So sin (pie/2)t v=S0 cos (pie/2)t *(pie/2)

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Kinematics

take y direction in direction of throw

\time

T = 2s~

[4]

Q.12 A large number of bullets are fired in all direction with the same speed v. What is the maximum area on ground on which these bullets can spread? Q.13 A boat starts from rest from one end of a bank of a river of width d flowing with velocity u. The boat is steered with constant acceleration a in a direction perpendicular to the bank. Ifpoint of start is origin, direction ofbank is x axis and perpendicular to bank is y axis. Find the equation oftrajectory ofthe boat. Q.14 A ball is thrown horizontallyfroma cliff such that it strikes ground after 5 sec. The line of sightfromthe point of projection to the point ofhitting makes an angle of 37° with the horizontal. What is the initial velocity ofprojection. r=v^2/g

Q.15 A ball is proj ected on smooth inclined plane in direction perpendicular to line of greatest slope with velocity of 8m/s. Find it's speed after 1 sec.

8 m/s

its a 3-d motion if see from ground we take inclined plane as refrence axis a=g cos 53

Q.16 A glass wind screen whose inclination with the vertical can be changed, is mounted on a cart as shown infigure.The cart moves uniformly along the horizontal path with a speed of 6 m/s. At what maximum angle a to the vertical can the wind screen be placed so that the rain drops falling vertically downwards with velocity 2 m/s, do not enter the cart?

o

o

mmn777777777777777777/7777777777

solution is given on next page

Q.17 A particle is proj ectedfrompoint P with velocity 5 A/2 m/s perpendicular to the surface of a hollowrightangle cone whose axis is vertical. It collides at Q normally. Find the time ofthe flight ofthe particle.

y

t--

thrown 45 degree from ground and received at 45 degree vy=5 in start and -5 at end so 1 sec

Q.18 Find range ofproj ectile on the inclined plane which is proj ected perpendicular to the incline plane with velocity 20m/s as shown in figure.

u = 20ms-'

alpha=53

37°X C- P«—X- -D A; 1.5m -B

Q.19 AB and CD are two smooth parallel walls. A child rolls a ball along ground from A towards point P find PD so that ball reaches point B after striking the wall CD. Given coefficient of restitution e = 0.5

let initially v= a i+ b j then after strick v= a i + b/2 j. time to go from A to P is half time from P to B but in x v is constant. so 2*CP=PD but CP+PD=1.5

Q.20 Initial acceleration of a particle moving in a straight line is a and initial velocity is zero. The acceleration reduces continuously to half in every t seconds as a =a . Find the terminal velocity of the particle. 2— ta0 mvuuuuummuwmv Q.21 Find the acceleration of movable pulley P and block B if rH rK acceleration of block A = 1 m/s 4-. 0

Q

calculate v by definite integration i.e. 0 to t then t tends to infinity

2

El

)£

m

^777777777777777777777777

Q.22 The velocities of Aand B are marked inthefigure.Find the velocity of block C (assume that the pulleys are ideal and string inextensible). by constrained motion velocity of c= 4 down but c is also moving with B at 3 so 3^2+4^2=25

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Kinematics

3m/s lm/s

J3

B

777777777777777777777777

[5]

Q.23 A particle is moving in x-y plane such that x = t + sin(t) meter, y = cos (t) meter, t is the time in sec. Find the length of the path taken by the particle from t = 0 to t = 2n sec.

calculate vx and vy then find v=root (vx^2+vy^2) then integrate it from 0 to pie and pie to 2 pie separately

Q.24 The speed of a particle when it is at its greatest height ^2/5 is of its speed when it is at its half the maximum height. The angle ofproj ection is and the velocity vector angle at half the maximum height is .

straight part of rope will increase with a slant part of rope will decrease by a (-a cos alpha i +a sin alpha j) so block will move in x =a- a cos alpha y=a sin alpha

Q.25 A weightless inextensible rope on a stationary wedge forming angle a with the horizontal. One end of the rope is fixed to the wall at point A. A small load is attached to the rope at point B. The wedge starts moving to therightwith a constant acceleration. Determine the acceleration a, ofthe load when it is still on the wedge.

777777777777/

Q.26 The horizontal range of a projectiles is R and the maximum height attained by it is H. A strong wind now begins to blow in the direction of motion of the projectile, giving it a constant horizontal acceleration = g/2. Under the same conditions ofproj ection, find the horizontal range of the proj ectile. Q .27 Consider the acceleration of a particle for a given time't' at 'a' m/s followed immediately by retardation at the same rate of'a' m/s for time 't/2', as one cycle. If the particle startedfromrest,findthe distance travelled by it after 'n' such cycles in succession. 2

2

Q. 2 8 A particle is thrown horizontally with relative velocity 10 m/sfroman inclined plane, which is also moving with acceleration 10 m/s vertically upward. Find the time after which it lands on the plane (g = 10 m/s ) ^

10m/s

2

2

2

3 0

replace g in formul by a=20 relative acceleration

Now solve it for alpha

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Kinematics

[6]

EXERCISE # III

Q. 1 A steel ball bearing is releasedfromthe roof of a building. An observer standing infrontof a window 120 cm high observes that the ball takes 0.125 sec to fall from top to the bottom of the window. The ball continutes to fall & makes a completely elastic collision with side walk & reappears at the bottom of the window 2 s after passing it on the way down. How tall is the building ? Q. 2 A train takes 2 minutes to acquire its full speed 60kmphfromrest and 1 minute to come to restfromthe full speed. If somewhere in between two stations 1 km ofthe track be under repair and the limited speed on this part be fixed to 20kmph, find the late running ofthe train on account of this repair work, assuming otherwise normal at running of the train between the stations. Q. 3 A speeder in an automobile passes a stationary policeman who is hiding behind a bill board with a motorcycle. After a 2.0 sec delay (reaction time) the policeman accelerates to his maximum speed of 150 km/hr in 12 sec and catches the speeder 1.5 km beyond the billboard. Find the speed of speeder in km/hr.

get u at top of window by 2nd eq 1.2=u*.125+5*.125*.125 u=8.975 T=1+.125+.8975=2.0225s h=5*2.0225*2.0225=20.4525

d2=40*2/180=4/9 km d1=40*4/180=8/9 km -(12/9)/60+(12/9)/40-1/60+1/20=2/45 hr =8/3 min =160sec

here policeman will travel with uniform velocity after t=2+12=14s

Q. 4

Aballoon is ascending vertically with an acceleration of 0.2m/s , Two stones are droppedfromit at an interval of 2 sec. Find the distance between them 1.5 sec after the second stone is released.(use g=9.8m/s )

Q.5

A ship steaming north at the rate of 12 km/h observes a ship due east to itself and distant 10 km, which steaming due west at the rate of 16 km/h. After what time they are at least distancefromone another and what is this least distance. An aeroplane is observed by two persons travelling at 60 km/hr in two vehicles moving in opposite directions on a straight road. To an observer in one vehicle the plane appears to cross the road track at right angles while to the observer in the other vehicle the angle appears to be 45°. At what angle does the plane actually cross the road track and what is its speed relative to the ground. A girl can paddle her canoe at 5m/sec. in still water. She wishes to cross a straight river which is flowing at 3m/sec. At what angle to the river bank should she steer to cross, (a) as quickly as possible, (b) by the shortest route. How long will aplane take to fly around a square with side a with the wind blowing at a velocity u, in the two cases the direction ofthe wind coincides with one ofthe sides the direction ofthe wind coincides with one diagonal ofthe square. The velocity ofthe plane in still air is v > u. 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 constantly heads for ship A, having at each moment the same speed as the latter. After a sufficiently great interval oftime the second ship will obviously follow thefirstone at a certain distance. Find the distance. The slopes of the wind-screen of two motorcars are p = 3 0° and p = 15° respectively. The first car is travelling with a velocity of v horizontally. The second car is travelling with a velocity v in the same direction. The hail stones are falling vertically. Both the drivers observe that the hail stones rebound vertically after elastic collision with the wind-screen. Find the ratio of v,/v A rocket is launched at an angle 53° to the horizontal with an initial speed of 100 ms . It moves along its initial line of motion with an acceleration of 30 ms~ for 3 seconds. At this time its engine falls & the rocket proceeds like afreebody. Find : the maximum altitude reached by the rocket total time of flight. the horizontal range . [ sin 53° = 4/5 ]

2

let u is velocity of ballon when1st droped in 2 sec w.r.t. ground v=u-(9.8)2 speed of second v2=u+.4 distance b/w them when 2nd dropped (w.r.t. ballon) d=1/2(9.8+.2)*4=20 Now with respect to 1st speed of 2nd=20

2

it is from still water not from 12 km/h ship

Q. 6 Q.7 Q. 8 (a) (b) Q. 9

direction of B changing regularly toward A ake A as reference frame than B will move at 45 from AB line, least distance between A and B is our answer

Q. 10

2

t

2

r

Q. 11 (i) (ii) (iii)

^Bansal Classes

_1

2

Kinematics

[7]

Q.12 A small ball is thrown between two vertical walls such that in the absence of the wall its range would have been 5d. The angle of projection is a. Given that all the collisions are perfectly elastic, find (a) Maximum height attained by the ball. \u\uu\uuvwu\ (b) Total number of collisions before the ball comes back to the ground, and d/2 (c) Point at which the ball fallsfinally.The walls are supposed to be very tall. Q.13 A hunter is riding an elephant ofheight 4m moving in straight line with uniform speed of 2m/sec. A deer running with a speed V infrontat a distance of 4V5m moving perpendicular to the direction of motion of the elephant. If hunter can throw his spear with a speed of 1 Om/sec. relative to the elephant, then at what angle 0 to it's direction of motion must he throw his spear horizontally for a successful hit. Find also the speed 'V' ofthe deer. Q.14 A perfectly elastic ball is thrownfromthe foot of a smooth plane inclined at an angle a to the horizontal. If after striking the plane at a distance Ifromthe point of projection, it rebounds and retraces its former gl (1 + 3 sin a) path, show that the velocity of projection is 2 sin a Q.15 A particle is proj ectedfromthe foot of an inclined plane at an angle a in the vertical plane through the line of greatest slope & hits the plane at right angles. If p be the angle the direction of projection makes with the plane & if the particle returns to the point of proj ection in two jumps,findthe value ofthe coefficient ofrestitution. Q.16 A projectile is to be thrown horizontallyfromthe top of a wall of height 1.7 m. Calculate the initial velocity ofprojection if it hits perpendicularly an incline of angle 37° which startsfromthe ground at the bottom of the wall. The line of greatest slope of incline lies in the plane ofmotion of projectile. Q.17 Two inclined planes OA and OB having inclination (with horizontal) 30° and 60° respectively, intersect each other at O as shown infig.Aparticle is projected from point P with velocity u = \ 0^3 m s along a direction perpendicular to plane OA. Ifthe particle strikes plane OB perpendicularly at Q, calculate velocity with which particle strikes the plane OB, (a) (b) time offlight, (c) vertical height h of PfromO, (d) maximum heightfromO attained by the particle and (e) distance PQ Q.18 A particle is projected with a velocity 2 ^/ag 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 2-JaJg • Q.19 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 ofthe bird and the stone, if the stone still hits the bird while descending. Q.20 Two persons Ram and Shyam are throwing ball at each other as shown in thefigure.The maximum horizontal distancefromthe building where Ram can stand and still throw a ball at Shyam is dj. The maximum ^ horizontal distance of Ramfromthe building where Shyam can throw a ball is d . If both of them can throw ball with a velocity of ^2gk, find mm' m u u u m m fc-nn ufl m the ratio of dj/d . Neglect the height of each person. 2

_1

Shyam

2

2

faBansal Classes

Kinematics

[8]

EXERCISE # III Q. 1

The motion of a body is given by the equation = 6 . 0 - 3 v(t) ; where v (t) is the speed in m/s & t in sec., if the body has v = 0 at t = 0 then (A) the terminal speed is 2.0 m/s (B) the magnitude of the initial acceleration is 6.0 m/s (C) the speed varies with time as v(t) = 2(l -e~ )m/s (D) the speed is 1.0 m/s when the acceleration is halfthe initial value. [JEE' 1995] 2

3t

Q.2

Two guns, situated at the top of a hill of height 10 m, fire one shot each with the same speed 5 yfs m/s at; some interval oftime. One gun fires horizontally and other fires upwards at an angle of 60° with the horizontal. The shots collide in air at a point P. Find (a) the time interval between thefirings,and (b) the coordinates ofthe point P. Take origin of the coordinates system at the foot ofthe hill right below the muzzle and traj ectories in X-Y plane. [JEE' 1996]

Q. 3

The traj ectory ofa proj ectile in a vertical plane is y = ax - bx , where a, b are constants & x and y are respectively the horizontal & vertical distances ofthe projectilefromthe point ofprojection. The maximum height attained is & the angle of projectionfromthe horizontal is . [JEE' 1997]

Q.4

A large heavy box is sliding without friction down a smooth plane of inclination 9. From a point P on the bottom ofa box, a particle is proj ected inside the box. The'initial speed ofthe particle with respect to box is u and the direction of projection makes an angle a with the bottom as shown in figure. i-

(a) (b) Q.5

2

the particle lands. (Assume that the particle does not litany other surface of the box. Neglect air resistance). , ' Ifthe horizontal displacement ofthe particle as seen by an observer on the ground is zero,findthe speed of the box with respect to the ground at the instant when the particle was projected. [JEE' 1998] A particle of mass 10~ kg is moving slong the positive x-axis under the influence of a force —K F(x)= whereK= 10 Nm .Attimet = 0itisatx-1.0m&itsvelocityisv = 0. Find: 2x its velocity when itit reaches the time at which reaches xx == 0.5 0.250 mm. [JEE' 1998] In 1.0 sec. a particle goesfrompoint Ato point B moving in a semicircle ofradius 1.0 m. The magnitude of average velocity is: [JEE '99] (A) 3.14 m/sec (B) 2.0 m/sec im (C) 1.0 m/sec (D) zero BThe co-ordinates of a particle moving in a plane are given by x (t) = a cos (7it) and y (t) = b sin (rat) where a, b (