1. A particle is moving along x-axis with its position (x) varying with time (t) as x = αt⁴ + βt² + γt + δ. The ratio of its initial velocity to its initial acceleration, respectively, is:

(1).2α : δ

(2). γ : 2δ

(3). 4α : β

(4). γ : 2β

2. The velocity (v)− time (t) plot of the motion of a body is shown below:

The acceleration (a)− time (t) graph that best suits this motion is :

(1).

(2).

(3).

(4).

3. The position of a particle is given by\(\displaystyle \vec{r}\left(t\right) = 4t\hat{i} +2t^2\hat{j}+5\hat{k}\) where t is in seconds and r in metre. Find the magnitude and direction of velocity v(t), at t = 1 s, with respect to x-axis

(1).4√2ms⁻¹, 45^∘

(2). 4√2ms⁻¹, 60^∘

(3). 3√2ms⁻¹, 30^∘

(4). 3√2ms⁻¹, 45^∘

4. A vehicle travels half the distance with speed v and the remaining distance with speed 2v. Its average speed is

(1).\(\displaystyle \frac{2v}{3}\)

(2). \(\displaystyle \frac{4v}{3}\)

(3). \(\displaystyle \frac{3v}{4}\)

(4). \(\displaystyle \frac{v}{3}\)

5. A bullet from a gun is fired on a rectangular wooden block with velocity u. When bullet travels 24cm through the block along its length horizontally, velocity of bullet becomes u/3. Then it further penetratesinto the block in the same direction before coming to rest exactly at the other end of the block. The total length of the block is

(1).24cm

(2). 28cm

(3). 30cm

(4). 27cm

6. A horizontal bridge is built across a river. A student standing on the bridge throws a small ball vertically upwards with a velocity 4ms⁻¹. The ball strikes the water surface after 4 s. The height of bridge above water surface is (. Take g = 10m s⁻²)

(1).60m

(2). 64m

(3). 68m

(4). 56m

7. The ratio of the distances travelled by a freely falling body in the 1 st , 2 nd , 3 rd and 4 th second

(1).1 : 2 : 3 : 4

(2). 1 : 4 : 9 : 16

(3). 1 : 3 : 5 : 7

(4). 1 : 1 : 1 : 1

8. The displacement-time graphs of two moving particles make angles of 30^∘ and 45^∘ with the x-axis as shown in the figure. The ratio of their respective velocity is

(1).√3 : 1

(2). 1:1

(3). 1:2

(4). 1 : √3

9. The position-time (x–t) graph for positive acceleration is :

(1).

(2).

(3).

(4).

10. A small block slides down on a smooth inclined plane, starting from rest at time t = 0. Let S_n be the distance travelled by the block in the interval t = n − 1 to t = n. Then, the ratio S_n/S_{n+ 1} is

(1).2n − 1/2n

(2). 2n − 1/2n + 1

(3). 2n + 1/2n − 1

(4). 2n/2n − 1

11. A car starts from rest and accelerates at 5 m/s². At t = 4 s, a ball is dropped out of a window by a person sitting in the car. What is the velocity and acceleration of the ball at t = 6 s?(Take g = 10 m/s²)

(1).20 m/s, 5 m/s²

(2). 20 m/s, 0

(3). 20√2 m/s, 0

(4). 20√2 m/s, 10 m/s²

12. A ball is thrown vertically downward with a velocity of 20m ∕ s from the top of a tower. It hits the ground after some time with a velocity of 80m ∕ s The height of the tower is : (g = 10m ∕ s²)

(1).340m

(2). 320m

(3). 300m

(4). 360m

13. Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time t₁. On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time t₂. The time taken by her to walk up on the moving escalator will be

(1).(t₁t₂)/ (t₂ -t₁)

(2). (t₁t₂)/ (t₂ + t₁)

(3). t₁ − t₂

(4). (t₁ + t₂)/2

14. If the velocity of a particle is v = At + Bt² , where A and B are constants,then the distance travelled by it between 1 s and 2 s is

(1).(3/2)A +(7/3)B

(2). A/2+B/3

(3). 3/2A + 4B

(4). 3A + 7B

15. Two cars P and Q start from a poin t at the same time in a straight line and their positions are represented by x_p(t) = (at + bt²) and x_Q(t) = (f t − t²) At what time do the cars have the same velocity ?

(1).(a − f)/ (1 + b)

(2). (a + f)/ 2(b − 1)

(3). (a + f)/ 2(1 + b)

(4). (f − a)/2(1 + b)

16. A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to v(x) = βx⁻²ⁿ, where β and n are constants and x is the position of the particle. The acceleration of the particle as a function of x is given by

(1).−2β²x^{−2n + 1}

(2). −2nβ²e^{−4n + 1}

(3). −2nβ²x^{−2n − 1}

(4). −2nβ²x^{−4n − 1}

17. A man throws balls with the same speed vertically upwards one after the other at an interval of 2 seconds. What should be the speed of the throw so that more than two balls are in the sky at any time? (Given g = 9.8m ∕ s² )

(1).more than 19.6 m/s

(2). at least 9.8 m/s

(3). any speed less than 19.6 m/s

(4). only with speed 19.6 m/s

18. If a ball is thrown vertically upwards with speed u, the distance covered during the last t seconds of its ascent is

(1).ut

(2). 1/2gt²

(3). ut −1/2 gt²

(4). (u + gt)t

19. A particle is thrown vertically upward. Its velocity at half of the height is 10m ∕ s then the maximum height attained by it (g = 10m ∕ s²)

(1).8m

(2). 20m

(3). 10m

(4). 16m

20. Motion of a particle is given by equation s = (3t³ + 7t² + 14t + 8)m. The value of acceleration of the particle at t = 1 sec is

(1).10m ∕ s²

(2). 32m ∕ s²

(3). 23m ∕ s²

(4). 16m ∕ s²

21. A car moving with a speed of 40km ∕ h can be stopped by applying brakes after atleast 2 m. If the same car is moving with a speed of 80km ∕ h, what is the minimum stopping distance?

(1).4m

(2). 6m

(3). 8m

(4). 2m

22. A rubber ball is dropped from a height of 5m on a plane. On bouncing it rises to 1.8m. The ball loses its velocity on bouncing by a factor of

(1).3/5

(2). 2/5

(3). 16/25

(4). 9/25

23. The position x of a particle varies with time, (t) as x = at² − bt³. The acceleration will be zero at time t is equal to

(1).a/3b

(2). 0

(3). 2a/3b

(4). a/b

24. If a car at rest accelerates uniformly to a speed of 144 km ∕ h in 20 sec, it covers a distance of

(1).1440cm

(2). 2980cm

(3). 20m

(4). 400m

25. A body dropped from a height h with initial velocity zero, strikes the ground with a velocity 3m ∕ s. Another body of same mass dropped from the same height h with an initial velocity of 4m ∕ s. The final velocity of second mass, with which it strikes the ground is

(1).5m ∕ s

(2). 12m ∕ s

(3). 3m ∕ s

(4). 4m ∕ s

26. The acceleration of a particle is increasing linearly with time t as bt. The particle starts from origin with an initial velocity v₀. The distance travelled by the particle in time t will be

(1).v₀t +1/3bt²

(2). v₀t +1/2bt²

(3). v₀t +1/6bt³

(4). v₀t +1/3bt³

27. The water drop falls at regular intervals from a tap 5m above the ground. The third drop is leaving the tap at instant the first drop touches the ground. How far above the ground is the second drop at that instant?

(1).3.75m

(2). 4.00m

(3). 1.25m

(4). 2.50m

28. A car accelerates from rest at a constant rate α for some time after which it decelerates at a constant rate β and comes to rest. If total time elapsed is t, then maximum velocity acquired by car will be

(1).(α² − β²)t/αβ

(2). (α² + β²)t/αβ

(3). (α + β)t/αβ

(4). αβt/α + β

29. 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

(1).3 m/s

(2). 42 m/s

(3). −15 m/s

(4). −9 m/s

30. The velocity of train increases uniformly from 20 km ∕ h to 60 km ∕ h in 4 hours. The distance travelled by the train during this period is

(1).160 km

(2). 180 km

(3). 100 km

(4). 120 km

31. The displacement-time graph of a moving particle is shown below.

The instantaneous velocity of the particle is negative at the point

(1).E

(2). F

(3). C

(4). D

32. A body starts from rest, what is the ratio of the distance travelled by the body during the 4th and 3rd second?

(1).7/5

(2). 5/7

(3). 7/3

(4). 3/7

33. Which of the following curve does not represent motion in one dimension?

(1).

(2).

(3).

(4).

34. A body dropped from top of a tower fall through 40m during the last two seconds of its fall. The height of tower is (g = 10m ∕ s²)

(1).60m

(2). 45m

(3). 80m

(4). 50m

35. A car moves a distance of 200m. It covers the first half of the distance at speed 40 km ∕ h and the second half of distance at speed v. The average speed is 48 km ∕ h. The value of v is

(1).56 km ∕ h

(2). 60 km ∕ h

(3). 50 km ∕ h

(4). 48 km ∕ h

36. A bus travelling the first one-third distance at a speed of 10 km ∕ h, the next one-third at 20 km ∕ h and at last one-third at 60 km ∕ h. The average speed of the bus is

(1).9 km ∕ h

(2). 16 km ∕ h

(3). 18 km ∕ h

(4). 48 km ∕ h

37. A car covers the first half of the distance between two places at 40 km ∕ h and another half at 60 km ∕ h. The average speed of the car is

(1).40 km ∕ h

(2). 48 km ∕ h

(3). 50 km ∕ h

(4). 60 km ∕ h

38. What will be the ratio of the distance moved by a freely falling body from rest in 4th and 5th seconds of journey?

(1).4:5

(2). 7:9

(3). 16:25

(4). 1:1

39. A car is moving along a straight road with a uniform acceleration. It passes through two points P and Q separated by a distance with velocity 30 km ∕ h and 40 km ∕ h respectively. The velocity of the car midway between P and Q is

(1).33.3 km ∕ h

(2). 20√2 km ∕ h

(3). 25√2 km ∕ h

(4). 35 km ∕ h

40. A ship A is moving westwards with a speed of 10 km\(h^{-1}\) and a ship B 100 km south of A, is moving northwards with a speed of 10 km\(h^{-1}\). The time after which the distance between them becomes shortest, is

(1). 5\(\sqrt{2}\) h

(2). 10\(\sqrt{2}\) h

(3). 0

(4). 5 h

41. The width of river is 1 km. The velocity of boat is 5 km / hr. The boat covered the width of river in shortest time 15 min. Then the velocity of river stream is

(1).3 km / hr

(2). 4 km / hr

(3). \(\sqrt{29}\) km / hr

(4). \(\sqrt{41}\) km / hr

42. A boat is sent across a river with a velocity of \(8\) \( km h^{-1}\). If the resultant velocity of boat is \(10\) \(km h^{-1}\), then velocity of river is

(1).\(12.8\) \( km h^{-1}\)

(2). \(6\) \( km h^{-1}\)

(3). \(8\) \( km h^{-1}\)

(4). \(10\) \( km h^{-1}\)

43. A train of 150 metre length is going towards north direction at a speed of 10m ∕ s. A parrot flies at the speed of 5m ∕ s towards south direction parallel to the railways track. The time taken by the parrot to cross the train is

(1).12 s

(2). 8 s

(3). 15 s

(4). 10 s

44. In some appropriate units, time (t) and position (x) relation of a moving particle is given by \(\text{t}\,=\,\text{x}^2\,+\,\text{x}\). The acceleration of the particle is

(1).\(\displaystyle + \frac{2}{2\text{x}+1}\)

(2). \(\displaystyle - \frac{2}{\left(\text{x}+2 \right)^3}\)

(3). \(\displaystyle - \frac{2}{\left(2\text{x}+1 \right)^3}\)

(4). \(\displaystyle + \frac{2}{\left(\text{x}+1 \right)^3}\)

45. Two cities X and Y are connected by a regular bus service with a bus leaving in either direction every T min. A girl is driving scooty with a speed of 60 km/h in the direction X to Y notices that a bus goes past her every 30 minutes in the direction of her motion, and every 10 minutes in the opposite direction, Choose the correct option for the period T of the bus service and the speed (assumed constant) of the buses.

(1).15 min, 120 km/h

(2). 9 min, 40 km/h

(3). 25 min, 100 km/h

(4). 10 min, 90 km/h