Work on paper, not in your head. Draw before you calculate — every question here rewards it, and several are much harder without a diagram.

1.  A particle moves along a straight line such that its displacement x (in meters) as a function of time t (in seconds) is given by x(t) = 2t³ − 9t² + 12t + 4. At what time t > 0 is the particle momentarily at rest?

(A)  t = 1 s and t = 2 s

(B)  t = 2 s and t = 3 s

(C)  t = 1.5 s only

(D)  t = 3 s only

2.  The position vector of a particle moving in a plane is r( t )=( 3t^2i+ 4t^3j )~m. What is the magnitude of the velocity vector at t = 1 s?

(A)  5 m/s

(B)  12 m/s

(C)  6√(5)~m/s

(D)  6√(3)~m/s

3.  A car accelerates uniformly from rest to a speed of 36 m/s over a distance of 180 m. What is the total time taken for this acceleration?

(A)  5 s

(B)  10 s

(C)  12 s

(D)  15 s

4.  A body is thrown vertically upward with an initial velocity u. If g is the acceleration due to gravity, the ratio of the time taken to reach half the maximum height to the total time of flight is:

(A)  frac1 -frac1√(2)2

(B)  frac1√(2)

(C)  (√(2))/(- 1)√(2)

(D)  1 -frac1√(2)

5.  Under constant acceleration a, a particle covers distances s¹ and s² in consecutive equal time intervals of T seconds. Which relation correctly expresses a?

(A)  a =fracs_2-s_1T^2

(B)  a =fracs_2+s_12T^2

(C)  a =frac2( s_2-s_1 )T^2

(D)  a =fracs_2-s_12T^2

6.  A particle starts from rest at t = 0 and moves with an acceleration a=(4 − 2t) m/s². The maximum velocity attained by the particle is:

(A)  2 m/s

(B)  4 m/s

(C)  8 m/s

(D)  16 m/s

7.  A particle’s position–time relationship is t = √x + 3, where x is in metres and t is in seconds. The position of the particle when its velocity is zero is:

(A)  0 m

(B)  3 m

(C)  6 m

(D)  −3 m

8.  The velocity-time graph of a moving object is a straight line passing through (0, 0) and (4, 16). The average velocity of the object over the time interval t = 0 to t = 4 s is:

(A)  4 m/s

(B)  8 m/s

(C)  12 m/s

(D)  16 m/s

9.  A particle moves along a circular path of radius R = 5 m at a constant tangential speed of 10 m/s. What is the magnitude of its net acceleration vector?

(A)  0 m/s²

(B)  10 m/s²

(C)  20 m/s²

(D)  50 m/s²

10.  A projectile is launched from ground level at an angle θ=45^(∘) with initial speed v⁰. The radius of curvature of its trajectory at the highest point is:

(A)  fracv_0^2g

(B)  fracv_0^22g

(C)  fracv_0^2g√(2)

(D)  (2)/(v)_0^2g

11.  A ball is dropped from a tower of height H. At the same instant, another ball is thrown vertically upward from the base of the tower with velocity u =√(2gH). The two balls cross each other at a height above ground equal to:

(A)  fracH4

(B)  fracH2

(C)  frac3H4

(D)  frac2H3

12.  The relation between position x and velocity v for a particle is given by v² = 16 − 9x² in SI units. What is the acceleration of the particle when x = 1 m?

(A)  − 9 m/s²

(B)  − 18 m/s²

(C)  9 m/s²

(D)  − 4.5 m/s²

13.  A river of width w = 400 m flows with a speed v_(r) = 3 m/s. A swimmer who can swim at v_(s) = 5 m/s in still water wants to cross the river in the shortest possible time. What is the drift along the riverbank experienced by the swimmer during this crossing?

(A)  0 m

(B)  120 m

(C)  240 m

(D)  300 m

14.  Two particles A and B move with constant velocities v_A=( 3i+ 4j )~m/s and v_B=( – 2i+ 9j )~m/s. The velocity of A relative to B (v_AB) is:

(A)  ( 5i- 5j )~m/s

(B)  ( i+ 13j )~m/s

(C)  ( – 5i+ 5j )~m/s

(D)  ( 1i- 5j )~m/s

15.  A train 150 m long is moving north at 15 m/s. A bird flies south parallel to the track at 10 m/s. How long does it take the bird to cross the train completely?

(A)  6 s

(B)  10 s

(C)  15 s

(D)  30 s

16.  Two blocks A and B are connected by light inextensible strings in an arrangement whose downward displacements always satisfy 3y_A + 2y_B = constant. At a certain instant block A is moving downward at 6 m/s. The velocity of block B at that instant is:

(A)  9 m/s upward

(B)  4 m/s upward

(C)  9 m/s downward

(D)  4 m/s downward

17.  If displacement of a particle is given by x = asin (ωt) + bcos (ωt), the acceleration of the particle is proportional to:

(A)  √(x)

(B)  x

(C)  − x

(D)  x²

18.  A body travels a distance s¹ with speed v¹ and the remaining distance s² with speed v². The average speed over the total distance is:

(A)  fracv_1+v_22

(B)  fracs_1+s_2fracs_1v_1+fracs_2v_2

(C)  fracs_1v_1+s_2v_2s_1+s_2

(D)  √(v)_1v_2

19.  In a v-x graph, if the velocity decreases linearly from v⁰ at x = 0 to 0 at x=x⁰, the acceleration at x = 0 is:

(A)  -fracv_0^2x_0

(B)  -fracv_0x_0

(C)  fracv_0^22x_0

(D)  -fracv_0^22x_0

20.  A bullet fired into a wooden target loses half of its velocity after penetrating 3 cm. Assuming constant resistive deceleration, how much further will it penetrate before coming to rest?

(A)  1 cm

(B)  1.5 cm

(C)  2 cm

(D)  3 cm

21.  Galileo’s Law of Odd Numbers states that the distances traversed during equal successive intervals of time by a body falling freely from rest are in the ratio:

(A)  1 : 2 : 3 : 4 : …

(B)  1 : 4 : 9 : 16 : …

(C)  1 : 3 : 5 : 7 : …

(D)  1 : 1 : 1 : 1 : …

22.  A vector trajectory is given by r( t )=( Rcosωt )i+( Rsinωt )j. The angle between the velocity vector and acceleration vector at any instant t is:

(A)  0^(∘)

(B)  45^(∘)

(C)  90^(∘)

(D)  180^(∘)

23.  The velocity of a particle moving along the x-axis varies as v = k√(x), where k is a positive constant. The acceleration of the particle is:

(A)  Zero

(B)  frack^22

(C)  k²

(D)  2k²

24.  An elevator ascends with a constant upward acceleration a = 2 m/s². A coin is dropped inside the elevator from a height of 2 m relative to the elevator floor. Taking g = 10 m/s², the time taken for the coin to hit the floor is:

(A)  frac1√(3)~s

(B)  √((2))/(5)~s

(C)  frac1√(2)~s

(D)  1 s

25.  A particle undergoes circular motion in a plane. If the angular displacement is θ(t) = 2t³ − 6t² rad, the angular acceleration becomes zero at t=

(A)  0.5 s

(B)  1.0 s

(C)  1.5 s

(D)  2.0 s

26.  A projectile is launched from ground level at an angle θ to the horizontal. If the horizontal range R is equal to twice the maximum height H reached, then tan θ is:

(A)  1

(B)  2

(C)  4

(D)  0.5

27.  The ratio of the numerical value of average velocity to average speed of a body moving along any path is always:

(A)  Equal to 1

(B)  Greater than 1

(C)  Less than or equal to 1

(D)  Greater than or equal to 1

28.  An object starts moving from rest with a constant acceleration a. The displacement during the n^(th) second is given by:

(A)  fraca2( 2n – 1 )

(B)  fraca2( n^2- 1 )

(C)  an

(D)  fraca2( 2n + 1 )

29.  A stone thrown horizontally from the top of a tower of height 80 m lands 100 m away from the foot of the tower. Taking g = 10 m/s², the initial speed of launch was:

(A)  20 m/s

(B)  25 m/s

(C)  40 m/s

(D)  50 m/s

30.  The displacement-time (x-t) graph for two particles A and B are straight lines making angles of 30^(∘) and 60^(∘) with the time axis, respectively. The ratio of their velocities v_(A):v_(B) is:

(A)  1 : 3

(B)  1:√(3)

(C)  √(3):1

(D)  1 : 1

31.  A particle moves along a circular track of radius r = 10 m with a tangential speed increasing at a rate of a_(t) = 3 m/s². At the moment when its speed is v = 10 m/s, the magnitude of total acceleration is:

(A)  3 m/s²

(B)  10 m/s²

(C)  √(109)~m/s^2

(D)  13 m/s²

32.  A body is thrown vertically upward with speed u. It passes a point at height h at time intervals t¹ and t². Which of the following relations is correct?

(A)  t_1t_2=(2)/(h)g

(B)  t_1+t_2=frachg

(C)  t_1t_2=frach2g

(D)  t_1-t_2=√(frac2)hg

33.  The velocity vector of a particle is given by v( t )=( 4ti+ 3j )~m/s. The equation of its trajectory, assuming it starts from the origin, is:

(A)  y =frac38x^2

(B)  x =frac29y^2

(C)  y =frac34x

(D)  x =frac43y

34.  In two-dimensional motion, if a_(x)=constant and a_(y)=constant, the trajectory of the particle must be:

(A)  Always a straight line

(B)  Either a straight line or a parabola

(C)  Always a circle

(D)  Always an ellipse

35.  A particle moves in a circle of radius R such that its speed depends on distance s as v = c√(s), where c is a constant. The tangential acceleration a_(t) is:

(A)  fracc^22

(B)  c²s

(C)  fracc^2sR

(D)  fracc^22R

36.  A rain dropped vertically falls with speed v_(r) = 4 m/s. A man walks horizontally with speed v_(m) = 3 m/s. At what angle to the vertical must he hold his umbrella to keep off the rain?

(A)  tan ⁻ ¹(3/4)

(B)  tan ⁻ ¹(4/3)

(C)  sin ⁻ ¹(3/4)

(D)  cos ⁻ ¹(3/4)

37.  The displacement x of a particle varies with time t as x = ae^( − αt) + be^(βt), where a, b, α, β are positive constants. The velocity of the particle will:

(A)  Be independent of time

(B)  Decrease continuously to zero

(C)  Go on increasing with time

(D)  Drop to zero at t = α/β

38.  The acceleration-time graph of a particle starting from rest is a triangle of base 4 s and height 6 m/s². The final velocity acquired by the particle at t = 4 s is:

(A)  24 m/s

(B)  12 m/s

(C)  6 m/s

(D)  18 m/s

39.  A plane is flying horizontally at v = 100 m/s at an altitude of 500 m. A bomb released from it will strike the ground after a horizontal displacement of (g = 10 m/s²):

(A)  500 m

(B)  1000 m

(C)  1500 m

(D)  2000 m

40.  A block slides down a smooth inclined plane of length L inclined at 30^(∘) to the horizontal. Starting from rest at the top, the time taken to reach the bottom is (g = 10 m/s²):

(A)  √((2L))/(g)

(B)  2√((L))/(g)

(C)  √((4L))/(5g)

(D)  √((L))/(g)

If your answer isn’t there

No answers are given, deliberately — and no question tells you which part it comes from, because deciding that is most of the work. If one defeats you, K16 maps twelve physical questions to the tool each one needs. Working it out a second time teaches more than checking a key.

But if you have worked carefully and your answer matches none of the options, it could be our mistake rather than yours. Every question was checked before publication, and checking is not the same as being right.

Those are the questions worth staying with. Rework them, and argue them out with a friend or a teacher. Showing that none of four options can be right is harder physics than picking the one that is — it needs you to trust your own derivation rather than search a menu for something familiar. A student who can do that has understood the motion.

Then tell us, and we will look at it properly. If the question is wrong we will correct it and say so on this page.