Kinematics
- In some appropriate units, time (t) and position (x) relation of a moving particle is given by t = x^2 + x. The acceleration of the particle is Answer: 2. -2/(2x+1)³ Key idea: differentiate t(x) to get velocity, then use a=v dv/dx . dt/dx=2x+1⇒ v=dx/dt=1/2x+1 a=vdv/dx=1/2x+1·(-2/(2x+... 1
- 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… Answer: 4. 15 min, 120 km/h Key idea: buses in one direction are spaced vT apart. The girl meets them at the rate set by the relative speed.... 1
- 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 : Answer: 4. graph (4) Key idea: Acceleration is the slope of the v - t graph. Initially v=0 and flat: a=0 . Straight rise: constant positive... 1
- A particle moving with uniform speed in a circular path maintains : Answer: 1. varying velocity and varying acceleration. Key idea: Velocity and acceleration are vectors, so a change in direction counts as a... 1
- A bullet from a gun is fired on a rectangular wooden block with velocity u. When bullet travels 24 cm through the block along its length horizontally, velocity of bullet becomes u/3. Then it further penetrates into the block in the same direction bef… Answer: 2. 27 cm Assume a constant resistive force, so deceleration a is constant. Use v²=u²-2as . First 24 cm: (u/3)²=u²-2a(24)⇒ 48a=8u²/9⇒... 1
- A horizontal bridge is built across a river. A student standing on the bridge throws a small ball vertically upwards with a velocity 4 m s^-1. The ball strikes the water surface after 4 s. The height of bridge above water surface is (Take g=10 m s^-2… Answer: 4. 64 m Take upward as positive. The ball starts with u=+4 m/s, and its displacement at the water is -h after 4 s. s=ut-1/2gt²⇒ -h=4... 1
- A vehicle travels half the distance with speed v and the remaining distance with speed 2v. Its average speed is: Answer: 4. 4v/3 Average speed = total distance / total time. Let the total distance be 2s . t₁=s/v , t₂=s/2v , total time =3s/2v . v_avg=2s/... 1
- A bullet is fired from a gun at the speed of 280 m s^-1 in the direction 30^° above the horizontal. The maximum height attained by the bullet is (g=9.8 m s^-2,\ 30^°=0.5) : Answer: 4. 1000 m Maximum height: H=u²sin²θ/2g . H=(280)²(0.5)²/2×9.8=78400×0.25/19.6=1000 m. The answer is 1000 m. 1
- The angular acceleration of a body, moving along the circumference of a circle, is : Answer: 1. along the axis of rotation Angular acceleration vecα = dvecω/dt is a vector along the axis of rotation, like vecω . For rotation... 1
- A ball is projected with a velocity, 10\ ms^-1, at an angle of 60° with the vertical direction. Its speed at the highest point of its trajectory will be : Answer: 3. 5√(3)\ ms⁻¹ The angle is measured from the vertical, so the angle with the horizontal is 30^° . At the highest point only the hor... 1
- 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 : Answer: 1. 1 : √(3) On a displacement-time graph, velocity equals the slope, v=tanθ . v₁/v₂=tan30^°/tan45^°=1/sqrt3/1=1:sqrt3 . The 30 degre... 1
- The ratio of the distances travelled by a freely falling body in the 1^st, 2^nd, 3^rd and 4^th second : Answer: 4. 1 : 3 : 5 : 7 For a body falling from rest, the distance in the n th second is s_n=1/2g(2n-1) . So the distances in 1st, 2nd, 3rd... 1
- A particle moving in a circle of radius R with a uniform speed takes a time T to complete one revolution.If this particle were projected with the same speed at an angle 'θ' to the horizontal, the maximum height attained by it equals 4R. The angle of… Answer: 4. θ = sin⁻¹(2gT²/π²R)^(1/2) Key idea: speed v=dfrac2π RT , and the maximum height is H=v²sin²θ/2g . 4R=4π²R²/T²·sin²θ/2g⇒sin²θ=8gRT... 1
- A car starts from rest and accelerates at 5 m/s^2. 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/s^2) Answer: 4. 20√(2) m/s, 10 m/s ² Key idea: once the ball is dropped, its horizontal velocity stays equal to the car's velocity at that instan... 1
- 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_nS_n+1 is : Answer: 2. 2n-1/2n+1 Key idea: distance in the n-th second from rest is S_n=1/2a(2n-1) . S_n∝(2n-1) and S_(n+1)∝(2n+1) , so dfracS_nS_(n+1)=... 1
- A ball is thrown vertically downward with a velocity of 20 m/s from the top of a tower. It hits the ground after some time with a velocity of 80 m/s. The height of the tower is : (g = 10 m/s^2) Answer: 4. 300 m Use v²=u²+2gh (downward motion, g positive). h=v²-u²/2g=80²-20²/2×10=6400-400/20=300\ m Answer: option 4. 1
- The speed of a swimmer in still water is 20 m/s. The speed of river water is 10 m/s and is flowing due east. If he is standing on the south bank and wishes to cross the river along the shortest path, the angle at which he should make his strokes w.r.… Answer: 1. 30^° west To cross along the shortest path (straight north), the swimmer's velocity component along the river flow must cancel th... 1
- Two particles A and B are moving in uniform circular motion in concentric circles of radii r_A and r_B with speed v_A and v_B respectively. Their time period of rotation is the same. The ratio of angular speed of A to that of B will be : Answer: 4. 1 : 1 Angular speed ω=2π/T . Both particles have the same period T , so they have the same angular speed. ω_A:ω_B=1:1 . (Their li... 1
- When an object is shot from the bottom of a long smooth inclined plane kept at an angle 60^° with horizontal, it can travel a distance x_1 along the plane. But when the inclination is decreased to 30^° and the same object is shot with the same veloci… Answer: 3. 1:√(3) On a smooth incline the retardation is gsinθ . For the same launch speed v , the distance up the plane is x=v²/2gsinθ , so... 1
- The x and y coordinates of the particle at any time are x = 5t - 2t^2 and y = 10t respectively, where x and y are in meters and t in seconds. The acceleration of the particle at t = 2s is : Answer: 2. - 4 m/s ² Key idea: acceleration components are second derivatives of position. a_x=d²x/dt²=-4 m/s ² , a_y=d²y/dt²=0 . Accelerati... 1
- Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time t_1. On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time t_2. The time… Answer: 2. t₁t₂/t₂+t₁ Key idea: add speeds. Let the stair length be L . Her walking speed is L/t₁ , escalator speed is L/t₂ . Combined speed... 1
- Two cars P and Q start from a point at the same time in a straight line and their positions are represented by x_P(t)=at+bt^2 and x_Q(t)=ft-t^2. At what time do the cars have the same velocity? Answer: 3. f-a/2(1+b) Velocity is the derivative of position; equate the two velocities. v_P=a+2bt ; v_Q=f-2t . a+2bt=f-2t⇒ 2t(b+1)=f-a . t=... 1
- In the given figure, a=15 m/s^2 represents the total acceleration of a particle moving in the clockwise direction in a circle of radius R=2.5 m at a given instant of time. The speed of the particle is Answer: 2. 5.7 m/s The total acceleration has a radial (centripetal) part v²/R and a tangential part. From the figure, a makes 30^° with the... 1
- If the velocity of a particle is v = At + Bt^2, where A and B are constants, then the distance travelled by it between 1s and 2s is : Answer: 3. 3/2A + 7/3B Key idea: distance is the integral of speed over time (here v≥0 ). s=∫₁²(At+Bt²)dt=A4-1/2+B8-1/3=3/2A+7/3B . 1