Tiêu đề 3-motion-in-two-dimensions-.pdf ✅

Motion in Two Dimensions 1. A particle starts traveling on a circle with constant tangential acceleration. The angle between velocity vector and acceleration vector, at the moment when particle completes half the circular track, is : (A) tan−1 (2π) (B) tan−1 (π) (C) tan−1 (3π) (D) zero 2. In a two dimensional motion, instantaneous speed v0 is a positive constant. Then which of the following are necessarily true? (A) The acceleration of the particle is zero (B) The acceleration of the particle is bounded (C) The acceleration of the particle is necessarily in the plane of motion (D) The particle must be undergoing a uniform circular motion 3. Two particles are projected in air with speed v0 at angles θ1 and θ2 (both acute) to the horizontal, respectively. If the height reached by the first particle is greater than that of the second, then tick the right choices. (A) Angle of projection : q1 > q2 (B) Time of flight: T1 > T2 (C) Horizontal range : R1 > R2 (D) Total energy : U1 > U2 4. A particle moving in a circle centered at the origin in anticlockwise sense as shown. The position of the particle is given as r⃗ = R(cos ωtiˆ + sin ωtjˆ) where, ω is constant. Mark the correct statement. (A) The particle has a constant acceleration (B) The particle has a variable acceleration (C) The acceleration of the particle changes according to the rate of | da⃗⃗ dt | = Rω 3 (D) The speed of the particle changes according to the rate of d|v ̅| dt = 0 5. A ball is projected from origin with speed 20 m/s at an angle 30∘ with x-axis. The x- coordinate of the ball at the instant when the velocity of the ball becomes perpendicular to the velocity of projection will be : (A) 40√3m (B) 40 m (C) 20√3m (D) 20 m 6. A particle is describing uniform circular motion in the anti-clockwise sense such that its time period of revolution is T. At t = 0 the particle is observed to be at A. If θ1 be the angle between acceleration at t = T 4 and average velocity in the time interval 0 to T 4 and θ2 be the angle between acceleration at t = T 4 and the change in velocity in the time interval 0 to T 4 , then : (A) θ1 = 135∘ , θ2 = 45∘ (B) θ1 = 135∘ , θ2 = 135∘ (C) θ1 = 45∘ , θ2 = 135∘ (D) θ1 = 45∘ , θ2 = 45∘ 7. In the given figure, a smooth parabolic wire track lies in the vertical plane (x − y plane). The shape of track is defined by the equation y = ( x 2 a ) (where a is constant). A ring of mass m which can slide freely on the wire track, is placed at the position A(a, a). The track is rotated with constant angular speed ω such that there is no relative slipping between the ring and the track then ω is equal to : (A) √ g a (B) √ g 2a
(C) √ 2g a (D) (2) 1 4 ( g a ) 1 2 8. A particle slides down a frictionless parabolic (y = x 2 ) track (A − B − C) starting from rest at point A (in figure). Point B is at the vertex of parabola and point C is at a height less than that of point A. After C, the particle moves freely in air as a projectile. If the particle reaches highest point at P, then : (A) KE at P = KE at B (B) Height at P = Height at A (C) Total energy at P = total energy at A (D) Time of travel from A to B = Time of travel from B to P. 9. A particle is thrown with a speed u at angle θ to the horizontal. When the particle makes an angle φ with the horizontal, its speed changes to v. (A) v = ucos θ (B) v = ucos θ ⋅ cos φ (C) v = cos θ ⋅ sec φ (D) v = usec θ ⋅ cos φ 10. For a particle performing uniform circular motion, choose the correct statement(s) from the following: (A) Magnitude of particle velocity (speed) remains constant (B) Particle velocity remains directed perpendicular to radius vector (C) Direction of acceleration keeps changing as particle moves (D) Angular momentum is constant in magnitude but direction keeps changing 11. A cart moves with a constant speed along a horizontal circular path. From the cart, a particle is thrown up vertically with respect to the cart. (A) The particle will land somewhere on the circular path (B) The particle will land outside the circular path (C) The particle will follow an elliptical path (D) The particle will follow a parabolic path 12. Two particles are projected from the same point with the same speed, at different angles θ1 and θ2 to the horizontal. They have the same horizontal range. Their times of flight are t1 and t2 respectively : (A) θ1 + θ2 = 90∘ (B) t1 t2 = tan θ1 (C) t1 t2 = tan θ2 (D) t1 sin θ1 = t2 sin θ2 13. A particle is projected with a speed u. After 2 seconds of projection it is found to be marking an angle of 45∘ with the horizontal and 0 ∘ after 3sec : (A) Angle of projection is tan−1 (3) (B) Angle of projection is tan−1 (1/3) (C) Speed of projection = 30√2 m/s (D) Speed of projection is 30 m/s 14. A ladder placed on a smooth floor slips. If at a given instant the velocity with which the ladder is slipping is, v1 and the velocity of that part of ladder which is touching the wall is v2, then the velocity of the centre of the ladder at that instant is : (A) v1 (B) v2 (C) v1+v2 2 (D) √v1 2+v2 2 2 15. A very broad elevator is going up vertically with a constant acceleration of 2 m/s 2 . At the instant when its velocity is 4 m/s a ball is projected from the floor of the lift with a speed of 4 m/s relative to the floor at an elevation of 30∘ . The time taken by the ball to return the floor is : (g = 10 m/s 2 )
(A) 0.5 s (B) 0.33 s (C) 0.25 s (D) 1s 16. The horizontal range and maximum height attained by a projectile are R and H respectively. If a constant horizontal acceleration a = g/4 is imparted to the projectile due to wind, then its horizontal range and maximum height will be : (A) (R + H), H 2 (B) (R + H 2 ) , 2H (C) (R + 2H), H (D) (R + H), H 17. For a particle moving along a circular path the radial acceleration ar is proportional to t 2 (square of time). If az is tangential acceleration which of the following is independent of time : (A) ar ⋅ az (B) az (C) ar/az (D) (ar ) 2 az 18. With what minimum speed must a particle be projected from origin so that it is able to pass through a given point (30 m, 40 m ). Take g = 10 m/s 2 : (A) 60 m/s (B) 30 m/s (C) 50 m/s (D) 40 m/s 19. A projectile is thrown with a velocity of 10√2 m/s at an angle of 45∘ with horizontal. The interval between the moments when speed is √125 m/s is : (g = 10 m/s 2 ) (A) 1.0 s (B) 1.5 s (C) 2.0 s (D) 0.5 s 20. Two particles A and B are separated by a horizontal distance x. They are projected at the same instant towards each other with speeds u√3 and u at angle of projections 30∘ and 60∘ respectively figure. The time after which the horizontal distance between them becomes zero is : (A) x u (B) x 2u (C) 2x u (D) 4x u 21. A projectile moves from the ground such that its horizontal displacement is x = Kt and vertical displacement is y = Kt(1 − αt), where K and α are constants and t is time. Find out total time of flight (T) and maximum height attained (Ymax) : (A) T = α, Ymax = K 2α (B) T = 1 α , Ymax = 2K α (C) T = 1 α , Ymax = K 6α (D) T = 1 α , Ymax = K 4α 22. A projectile is given in an initial velocity of (i + 2j)m/s, where i is along the ground and j is along the vertical. If g = 10 m/s 2 , the equation of its trajectory is : (A) y = x − 5x 2 (B) y = 2x − 5x 2 (C) 4y = 2x − 5x 2 (D) 4y = 2x − 25x 2 23. A projectile is aimed at a mark on a horizontal plane through the point of projection and falls 6 m short when its elevation is 30∘ but overshoot the mark by 9 m when its elevation is 45∘ . The angle of elevation of projectile to hit the target on the horizontal plane : (A) sin−1 [ 3√3+4 5 ] (B) cos−1 [ 3√3+4 5 ] (C) 1 2 cos−1 [ 3√3+4 10 ] (D) 1 2 sin−1 [ 3√3+4 10 ] 24. In uniform circular motion where B is fixed ωAO ωAB = ang. velocity of A wrt. O ang. velocity of A wrt.B (A) 1 2 (B) 2 (C) 1 (D) None of these