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Speed, Velocity and Acceleration:

 

(i)                  Speed is related to distance and it is a scalar while velocity is related to displacement and it is a vector.

(ii)        For a moving body speed cannot have zero or negative values but velocity can have.

(iii)       The ratio of total distance covered by a particle in given time interval divided by the time interval is called Average Speed.

 

           

 

            (iv) Instantaneous speed at any instant t is defined as -

 

           

           

            The slope of the time - distance graph provides the value of instantaneous speed.

            (v)  The ratio of displacement in a given time interval divided by the time interval is called Average Velocity -

 

                  

 

 

            (vi) The magnitude of instantaneous velocity is always equal to the instantaneous speed.

           

(vii) For any making object, the average speed can never be zero or negative but average velocity can have.

           

(viii) If a moving particle covers, distances S1, S2, S3 ……….. with speed V1, V2, V3 ……………… respectively then average speed –

 

           

           

            Where n = number of distances traveled

 

            (ix) If a particle movies in time interval t1, t2, t3 …. tn with speed v1, v2, v3 ……. vn then average speed of particle –

 

           

 

            (x) If a particle moves along a straight line without changing its direction. Then magnitude of average velocity be equal to the magnitude of average speed. Otherwise

 

           

 

            (xi) When a particle moves with constant velocity, its average velocity, instantaneous velocity and its speed all are equal.

 

            (xii) If an object moving with constant speed then its velocity may be constant or may be vary.

 

Example: - In uniform circular motion speed of particle be constant but velocity does not be constant because direction of velocity of moving particle continuously changes.

           

            (xiii) If a particle moves with constant velocity then speed of the particle must be constant and particle moves or straight line without changing its direction.

 

            (xiv) The ratio change in velocity in a time interval with the time interval is called average acceleration –

 

           

 

            The instantaneous acceleration is given by

 

           

           

(xv) If velocity of moving particle given in terms of position vector then acceleration of the particle -

 

           

 

           

           

            (xvi) Speedometer of automobile measures instantaneous speed.

 

 

 

Questions Answer

 

Q1.       The acceleration of a particle starting from rest varies with time according to the relation –

            A= - rw2 Sinwt

            Find displacement of this particle at a time t.

 

Ans.     According to Questions –

 

            a = -rw2 Sinwt

 

            or dv/dt = -rw2 sinwt

 

            or dv = -rw2 Sinwt dt

           

           

 

           

 

            or v = rw Coswt

 

           

 

            or dx = rw Coswt dt

 

           

 

           

           

            x = r Sinwt

 

Q.2       An electron starting from rest has a velocity that increased linearly with time that is V = Kt, where K = 2 m/s2 Find distance covered in first 3 sec.

 

Ans.     According to questions –

            V = kt

 

           

 

            dx = kt.dt

           

           

 

           

            Hence, distance covered in 3 sec is

 

           

 

            x = 9m

 

Q3.       What is the retardation of a moving particle? If the relation between time and position is

                                    t= Ax2 + Bx.

 

Ans.     According to problem –

 

                                    t = Ax2 + Bx

            Now differentiate above equation –

 

           

 

           

 

            = A (2x) + BX1

 

 

           

 

            or v = (2Ax + B)-1

           

            Again differentiate above equations –

                       

           

 

           

 

           

 

           

            If retardation be a then –

 

           

 

           

 

 

           

 

Q4.       The displacement of the particle is zero at t = 0 and it is x at t = 1. It starts moving in the positive x – direction with velocity which varies –

            v = kΦx

            where K is a constant. Find the relation for the variation of velocity with time.

 

Ans.     According to problem –

            V= K Φx ……………………(1)

 

            or dx/dt = K X ½

 

            or dx/x1/2 = Kdt

 

            or  x1/2 dx =  Kdt

            Now integrate above equation –

 

                                   

 

           

 

           

 

 

           

 

                              

 

Q5.       The displacement x of a particle moving in one dimension, under the action of a constant is related to time‘t’ by equation –

 

            t = Φ x + 3

            where x is in metre and t in Seconds. Find the velocity and acceleration of the particle at t = 2 sec.

 

Ans.     According to problem –

            t = Φx + 3

            or Φx = t – 3

            or x = (t – 3)2 = t2 + 9 – 6t

            Hence velocity –

           

           

 

            v = 2t – 6

           

            Hence, velocity at t = 2 sec.

            V = 2 x 2 – 6

 

            V = - 2m/s

 

Q7.       If a particle travels the first half distance with speed v1 and second half distance with speed v2 Find average speed during Journey.

 

Ans.     Let total distance be 2S.

            K --------------------25----------------------->|

            K---------S---------->|<----------S---------->|

            A-----------------------|-------------------------B

                        t1,v1                                    t2,v2

 

           

 

                                                                       

 

                                                                       

 

                        Second Method: According to the point (viii)

 

                    

 

           

 

           

 

Q8.       If a particle travels with speed V1 during first half time interval and with V2 speed during Second half time interval. Find its average speed during its journey.

 

Ans.     |< ---------------------S1------------------>|<-------------------S2-------------------->|

                                   

t, V1                                                                                   t, V2

            A · ------------------------------------------|-------------------------------------------·B

 

           

           

 

 

           

 

           

 

Q9.       The distance traveled by a particle in time t is given by –

                        S= (2.5/s2)t2

 

            Find (a) The average speed of the particle during the time 0 to 5.0 Sec and (b) The instantaneous speed at t = 5 sec.

 

Ans.     (a) the distance traveled between time 0 to 5 sec is –

                        S = (2.5 m/s2) (5.0)2

                        S = 62.5 m

            Hence,

 

           

 

            (b)  

                       

            V = 5 t

            Hence, instantaneous speed at time t = 5 Sec

           

            V = 5 (5)

 

            V = 25 m/s

 

Q10.     Straight distance between a hotel and a railway station is 10 km but circular route is followed by a taxi covering 23 km in 28 minute what is velocity? Are they equal?    

 

Ans.     (a) Average Speed =

                                   

 

           

 

 

           

 

            Here average sped and magnitude of Average velocity is not equal

 

Q11.     Three particles have position at time t-

            XA = 2t + 7, XB = 3t2 + 2t + 6, Xc = 5t2 + 4t

            Which of them have uniform acceleration?

 

Ans.     (a)

                       

            Hence, particle moves with constant velocity and its acceleration be zero.

 

           

 

            Hence, particle moves with constant acceleration equal to 6 m/s2­.

 

           

 

            Here, acceleration depends on time, so it is not uniform.

 

Q12.     A bird flies due north at velocity 20m/s For 15 Sec. It rest for 5 Sec and then fillies due south at velocity 24m/s for 10 sec. For the whole trip find the average speed and magnitude for average velocity.

 

Ans.     (a)

           

           

 

           

 

           

 

           

 

           

 

Q13.     A table clock has its minute hand 4.5 cm long. Find the average velocity of the tip of minute hand.

(a)     between 12.00 a.m. to 12.30 p.m.

(b)    between 12.00 a.m. to 12.30 p.m.

 

Ans.     (a) The displacement of tip minute hand between 12.00 a.m. and 12.30 p.m. will be 2r = (2 x 4.5)

 

           

 

Hence, average velocity between 12.00 a.m. and 122.30 p.m. –

 

 

            (b) The displacement of tip of minute hand between 12.00 a.m. and 12.30 pm. Will be also same. Hence average velocity of tip of minute hand between 12.00 a.m. and 12.30 p.m. -

 

             

 

Q14.     The initial velocity of a particle is u (at = 0) and the acceleration f is given by at. What is the equation of velocity of particle at any time t?

Ans.     According to questions –

            F= at

           

           

 

            or dv = at dt

now integrate above can

 

           

 

           

 

           

 

           

 

 

EXERCISE – 2

 

Q1.       The numerical value of the ratio of velocity to speed is –

            (i) always less then one             (ii) always equal to one

            (iii) always more than one          (iv) equal to or less then one

 

Ans.     (iv)

 

Q2.       A particle moves with uniform velocity, which of the following statements about the motion of the particle is ture-

            (i) Its speed is zero

            (ii) Its acceleration is zero

            (iii) Its acceleration opposite to velocity

            (iv) Its speed may be variable

 

Ans. (ii)

 

Q3.       The magnitude of the displacement is equal to the distance covered in a given interval of time, if the particle –

            (i) moves with constant acceleration.

            (ii) moves with constant speed.

            (iii) moves with constant velocity.

            (iv) none of the above.

 

Ans.     (iii)

 

Q4.       A moving body is covering the distance directly proportional to the square of the time. The acceleration of the body is:

            (i) increasing                 (ii) decreasing

            (iii) zero                        (iv) constant

 

Ans.     (iv)

 

Q5.       The position X of a particle varies with time (t) as –

            x = at2 – bt3

            The acceleration at time t of the particle will be equal to zero, where t is equal to –

 

                                                                                                     

 

Ans.     (iii)

 

Q6.       A body covers one – third of the distance with velocity V1. The Second one-third of the distance with a velocity V2 and the remaining distance with a velocity V3 the average velocity is –

           

                                             

 

                                   

           

Ans.     (ii)

 

Q7.       A particle moving in a straight line covers half the distance with speed 3m/s. The other half of the distance is covered in two equal time intervals with speed of 4.5 m/s and 7.5 m/s respectively. The average speed of the particle during this motion is –

            (i) 4.0 m/s

            (ii) 5.0m/s

            (iii) 5.5 m/s

            (iv) 4.8 m/s

 

Ans.     (i)

 

Q8.       A train of 150 m length is going towards north direction at a speed of 10m/s. A parrot flied at a speed of 5m/s towards south direction parallel to the railway track.  The time taken by the parrot to cross the train is equal to:

            (i) 12 sec

            (ii) 8 sec

            (iii) 12 sec

            (iv) 10 sec

 

Ans.     (iv)

 

Q9.       The distance traveled by a particle is directly proportional to 1/t2, where t = time elapsed. What is the nature of motion:

(i) Increasing acceleration          (ii) decreasing acceleration        (iii) In creasing retardation          (iv) Decreasing retardation

 

Ans.     (iv)      

 

Q10.     A particle moves along a straight line such that its displacement at any time t is given by –

            s = t3 – 6t2 + 3t + 4 metres

            The velocity when the acceleration is zero is –

            (i) 3m/s

            (ii) -12 m/s

            (iii) 42 m/s

            (iv) -9 m/s

 

Ans.     (iv)

 

Q11.     The co-ordinates of a moving particle at time t are given by x = ct2 and y = bt2

            The speed of particle is given by –

            (i) 2t (c+ b)

           

           

 

           

 

Ans.     (iv)

 

Q12.     A particle is moving eastward with a velocity of 5m/s.In 10 Sec, the velocity changes to 5m/s northward. The average acceleration in this time is –

            (i) zero

           

           

 

           

 

           

Ans.     (ii)

 

Q13.     The relation between time t and distance x is

            t = a x2 + b x

            where a and b are constants. The retardation is

            (i) 2a v3

            (ii) 2 b v3

            (iii) 2 a b v3

            (iv) 2 b2v3

 

Ans.     (i)

 

Q14.     A point moves in a straight line so that its displacement x m at time t sec is given by x2 = 1 + t2. Its acceleration in m/s2 at any time t sec is –

 

           

 

Ans.     (iii)

 

Q15.     The displacement of am particle after time t is given by

           

            where b is a constant. What is the acceleration of the particle –

            (i) Ke-bt

           

(ii) –Ke-bt

           

           

 

           

 

Ans.     (ii)

 

Q16.     Two cars get closer by 8m every second while traveling in the opposite directions. They get closer by 0.8 m every second while traveling in the same direction. What are the speeds of the two cars-

            (i) 4m/s, 4.4 m/s            (ii) 4.4 m/s. 3.6 m/s

            (iii) 4m/s, 3.6 m/s           (iv) 4m/s, 3 m/s

 

Ans.     (ii)

 

Q17.     The velocity v and displacement r of a body are related as –

            v2 = kr

            where K is a constant. What will be the velocity after 1 second? Given that the displacement is zero at t = 0.

            (i) ΦKr

            (ii) Kr3/2

            (iii) K/2 r0

            (iv) none

 

Ans.     (iii)

 

Q18.     A particle’s position as a function of time is described as –

            y (t) = 2t2 + 3t + 4

            What is the average velocity of the particle from t = 0, t = 3 sec –

            (i) 3m/s             (ii) 6m/s            (iii) 9m/s            (iv) 19 sec

 

Ans.                

 

Q19.     The displacement of a particle starting from rest (at t = 0) is given by –

            S = 6t2 – t3

            The time in second at which at particle will attain zero velocity again is -

            (i) 4 sec                        (ii) 8 sec           (iii) 12 sec         (iv) 16 sec

 

Ans.     (i)

 

Q20.     Two particles A and B are connected by a rigid rod AB. The rod slides along the perpendicular rails as shown. The velocity of A to the left is 10m/s. What is the velocity of end B, If a = 200 and tan 200 = 0.363.

            (i) 5.78 m/s

            (ii) 9.8 m/s

            (iii) 11.2 m/s     

            (iv) 3.63 m/s

 

Ans.     (iv)

 

Q21.     Check up the only correct statement in the following-

            (i) A body has constant velocity and still it can have a varying speed.

            (ii) A body has constant speed but it can have a varying velocity.

            (iii) A body having constant speed cannot have any acceleration.

            (iv) A body in motion under force acting upon it must always have work done upon it.

 

Ans. (ii)

 

Q22.     Three persons are initially at the three corners of an equilateral triangle whose side is equal to d. Each person now moves with a uniform speed V in Such a way that the first moves directly towards the second, the second directly towards the third, the third directly towards the first. The three persons will meet after a time equal to-

           

 

           

 

           

 

           

 

Ans.     (ii)       

 

Q23.     The deceleration experienced by a moving motor boat, after its engine is cut off is given by dv/dt = -KV3, where K is constant. If V0 is the magnitude of the velocity at cut off, the magnitude of the velocity at a time t after the cut – off is –

           

           

(ii) V0

           

            (iii) V0e -kt

 

            (iv) none of these

 

Ans.     (iv)

 

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