Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

JEE: Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

The document Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Physics 35 Years JEE Main & Advanced Past year Papers.
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Q.1. A car accelerates from rest at a constant rate a for some time after which it decelerates at a constant rate b to come to rest. If the total time lapse is t seconds, evaluate. (1978) (i) maximum velocity reached, and (ii) the total distance travelled.

Ans.
Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

 Solution.
Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Distance travelled = area of Δ ABC

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.2.The displacement x of particle moving in one dimension, under the action of a constant force is related to the time t by the equation t =√x+ 3 where x is in meters and t in seconds. Find

(i) The displacement of the particle when its velocity is zero, and

(ii) The work done by the force in the first 6 seconds.

Ans.(i) 0 ;  (ii) 0

Solution. Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

(i) For velocity to be zero, 2 t – 6 = 0 ⇨ t = 3 sec.

The displacement is x = 9 + 9 – 6 × 3 = 0

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

At t = 6 sec, v = 6 ms–1

 Work done = Change in K.E. = [K.Ef  – K.Ei]

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.3. Answer the following giving reasons in brief :   Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Is the time variation of position, shown in the figure observed in nature?      (1979)

Ans. No

Solution. Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

As shown, at a given instant of time, the body is at two different positions A and B which is not possible.

Q.4. Particles P and Q of mass 20 gm and 40 gm respectively are simultaneously projected from points A and B on the ground.

The initial velocities of P and Q make 45° and 135° angles respectively with the horizontal AB as shown in  the figure.

Each particle has an initial speed of 49 m/s. The separation AB is 245 m.

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE


Both particle travel in the same vertical plane and undergo a collision After the collision, P retraces its path, Determine the position of Q when it hits the ground. How much time after the collision does the particle Q take to reach the ground? Take g = 9.8 m/s2.

Ans. mid point of AB, 3.53 sec.

Solution. If a body dr ops fr om a h eigh t H above the ground then the time taken by it to reach the ground

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.5. Two towers AB and CD are situated a distance d apart as shown in figure.

AB  is 20 m high and CD is 30  m high from the ground. An object of mass m is thrown from the top of AB horizontally with a velocity of 10 m/s towards CD. (1994 - 6 Marks) Simultaneously another object of mass 2 m is thrown from the top of CD at an angle of 60° to the horizontal towards AB with the same magnitude of initial velocity as that of the first object. The two objects move in the same vertical plane, collide in mid-air and stick to each other.  Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

 (i) Calculate the distance ‘d’ between the towers and,

 (ii) Find the position where the objects hit the ground.

Ans. 17.32, 11.547 m from B 

Solution.

(i) Let t be the time taken for collision.

For mass m thrown horizontally from A.

For horizontal motion PM = 10 t          ... (i) For vertical motion uy = 0; sy = y; ay = g; ty = t

∴ y = 1/ 2 gt2    ... (ii)

vy = uy + ay t = gt   ... (iii)

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

For mass 2m thrown from C 

For horizontal motion  QM = [10 cos 60°] t     ... (iv)

For vertical motion  vy = 10 sin 60° = 5√ 3 ; ay = g sy = y + 10; ty = t

Now, vy = 5√ 3 + gt  ... (v)

and (sy) = uyt + 1/2 at 2

⇒ y + 10 = 5√3 1/2 gt2         ... (vi)

From (ii) and (vi)

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

= 10√3 = 17.32 m

(ii) Applying conservation of linear momentum (during collision of the masses at M) in the horizontal direction

 m × 10 – 2 m 10 cos 60° = 3 m × vx

 10 m – 10 m = 3 m × vx ⇨ vx = 0

Since, the horizontal momentum comes out to be zero, the combination of masses will drop vertically downwards and fall at E.

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.6.Two guns, situated on the top of a hill of height 10 m, fire one shot each with the same speed 5 √3 m s–1 at some interval of time. 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 (i) the time-interval between the firings, and (ii) the coordinates of the point P. Take origin of the coordinate system at the foot of the hill right below the muzzle and trajectories in x-y plane. (1996 - 5 Marks)

Ans.1 sec, (5√3 , 5) in metres

Solution.

 For Bullet A. Let t be the time taken by bullet A to reach P.

Vertical motion

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

10 – y = 5 t2

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Horizontal motion x = 5√3 t ... (ii)

For bullet B.

Let (t + t') be the time taken by bullet B to reach P Vertical Motion Let us consider upward direction negative and downward positive. Then

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Horizontal motion

x = (5 √3 cos 60°) (t + t ')

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Substituting the value of x from (ii) in (iv), we get

5√3t + 5√3t '= 10√3t

⇒ t = t '

Putting t = t ' in eq. (iii) y – 10 = 15 t – 20 t2 ... (v)

Adding (i) and (v)

 0 = 15 t – 15 t2 ⇒ t = 1 sec.

(ii) Putting t = 1 in eq. (ii), we get x = 5√3

 Putting t = 1 in eq. (i), we get y = 5

Therefore, the coordinates of point P are ( 5√3 , 5) in metres.

Q.7.A large, heavy box is sliding without friction down a smooth plane of inclination q. From a point P on the bottom of the box, a particle is projected inside the box. The initial speed of the particle with respect to the box is u, and the direction of projection makes an angle a with the bottom as shown in Figure. (1998 - 8 Marks)

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE


(a) Find the distance along the bottom of the box between the point of projection P and the point Q where the particle lands. (Assume that the particle does not hit any other surface of the box. Neglect air resistance.) 

(b) If the horizontal displacement of the particle as seen by an observer on the ground is zero, find the speed of the box with respect to the ground at the instant when particle was projected.

Ans.
Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE
 


Solution. 

(a) u is the relative velocity of the particle with respect to the box. Resolve u. uis the relative velocity of particle with respect to the box in x-direction. uy is the relative velocity with respect to the box in y-direction.

Since, there is no velocity of the box in the y-direction, therefore this is the vertical velocity of the particle with respect to ground also.

Y-direction motion (Taking relative terms w.r.t. box)

uy = + u sinα

a= – g cos∅

sy = 0 (activity is taken till the time the particle comes back to the box.)

ty = t

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

X - direction motion (Taking relative terms w.r.t. box)

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

(b) For the observer (on ground) to see the horizontal displacement to be zero, the distance travelled by the box in timeSubjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE should be equal to the range of the particle.

Let the speed of the box at the time of projection of particle be U. Then for the motion of  box with respect to ground.

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

On solving we get  U =Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.8. An object A is kept fixed at the point x = 3 m 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 ur as shown. A stationary person on the ground observes the stone hitting the object during its downward motion at an angle of 45° to the horizontal. All the motions are in the X–Y plane. Find ur  and the time after which the stone hits the object. Take g = 10 m/s    (2000 - 10  Marks)


Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEEAns.Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. 

Let 't ' be the time after which the stone hits the object and θ be the angle which the velocity vector u  makes with horizontal.

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

According to question, we have following three conditions. (i) Vertical displacement of stone is 1.25 m.

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

where g = 10 m/s2

or (u sinθ ) t  = 1.25 + 5t2

(ii) Horizontal displacement of stone = 3 + displacement of object A.

Therefore, (u cos θ ) t= 3 + 1/2 at2

where a = 1.5 m/s2

or(u cos q )t = 3 + 0.75 t2     ... (ii)

 Horizontal component of velocity of stone = vertical component (because velocity vector is inclined at 45° with horizontal.) Therefore (u cos θ) = gt – (u sinθ ) ... (iii) 

(The right hand side is written gt – u sinθ because the stone is in its downward motion. Therefore, gt > u sinθ .

In upward motion usinθ > gt). Multiplying equation (iii) with t  we can write, (u cosθ ) t + (u sin q ) t = 10 t2    ... (iv)

Now, (iv) – (ii) – (i) gives 4.25 t2 – 4.25 = 0 or t = 1s Substituting t = 1s in (i) and (ii), we get, u sin q = 6.25 m/s or uy = 6.25 m/s and u cos q = 3.75 m/s.

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.9. On a frictionless horizontal surface, assumed to be th e x-y plane, a small trolley A is moving along a straight line parallel to the y-axis (see figure) with a constant velocity of (√ 3 -1) m/s. At a particular instant, when the line OA makes an angle of 45° with the x-axis, a ball is thrown along the surface from the origin O. Its velocity makes an angle f with the x-axis and it hits the trolley.

(a) The motion of the ball is observed from the frame of the trolley. Calculate the angle q made by the velocity vector of the ball with the x-axis in this frame.    

(b) Find the speed of the ball with respect to the surface, ifφ = 4θ/4.    (2002 - 5 Marks)

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. 45°, 2 m/sec.

Solution. (a) Let the ball strike the trolley at B. Let

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Velocity of ball w.r.t. trolley

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

From triangle OAB

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

From (i) and (ii) Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ velocity of ball w.r.t. trolley makes an angle of 45° with the X-axis

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

(b) Here q = 45°

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

In ΔOMA,

θ = 45°⇨∠OAM = 45°     

∴ ∠OAB = 135°

Also∠BOA = 60° – 45° = 15°

Using sine law in ΔOBA 

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

The document Subjective Type Questions: Motion | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Physics 35 Years JEE Main & Advanced Past year Papers.
All you need of JEE at this link: JEE

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