CBSE Class 11 Physics Kinematics Advanced Problems

1. A particle of mass 10^–2 kg is moving along the positive X-axis under the influence of a force F (x) = – k/2x² where k = 10^–2 Nm². At time t = 0, it is at x = 1.0 m and its velocity is v = 0.
(a) find the velocity when it reaches x = 0.5 m (b) find the time at which it reaches x = 0.25 m

2. A car starts moving rectilinearly, first with acceleration a = 5.0 m/s² (the initial velocity is equal to zero), then uniformly, and finally, decelerating at the same rate a, comes to a stop. The total time of motion equals τ = 25 s. The average velocity during that time is equal to = 72 km per hour. How long does the car move uniformly ?

3. At the moment t = 0 a particle leaves the origin and moves in the positive direction of the x axis. Its velocity varies with time as v = v0 (1 – t / τ), where v0 is the initial velocity vector whose modulus equals v0 = 10.0 cm/s ; τ = 5.0 s. Find :
(a) the x coordinate of the particle at the moments of time 6.0, 10 and 20 s
(b) the moments of time when the particle is at the distance 10.0 cm from the origin
(c) the distance s covered by the particle during the first 4.0 and 8.0 s

4. The velocity of a particle moving in the positive direction of the x axis varies as v = α √x , where α is a positive constant. Assuming that at the moment t = 0 the particle was located at the point x = 0, find :
(a) the time dependence of the velocity and the acceleration of the particle
(b) the mean velocity of the particle averaged over the time that the particle takes to cover the first s meters of the path.

5. A point moves rectilinearly with deceleration whose modulus depends on the velocity v of the particle as a = α √v , where α is a positive constant. At the initial moment the velocity of the point is equal to v0. What distance will it traverse before it stops ? What time will it take to cover that distance ?

6. The current velocity of a river grows in proportion to the distance from its bank and reaches the maximum value v0 in the middle. Near the banks the velocity is zero. A boat is moving along the river in such a manner that it is always perpendicular to the current and the speed of the boat in still water is u. Find the distance through which the boat crossing the river will be carried away by the current if the width of the river is c. Also determine the trajectory of the boat.

7. A particle moves in the plane xy with velocity v = ai + bxj, where i and j are the unit vectors of the x and y axes, and a and b are constants. At the initial moment of time the particle was located at the point x = y = 0. Find the equation of the particle’s trajectory y (x).

8. A particle of mass 1 kg which moves along the x-axis is subjected to an accelerating force which increases linearly with time and a retarding force which increases directly with displacement (constant of proportionality being one with proper dimensions is both the cases. At time t = 0, displacement and velocity both are zero. Find the displacement as a function of time t.

9. A car starts from rest with an acceleration of 6 m/s² which decreases to zero linearly with time, in 10 second, after which the car continues at a constant speed. Find the time required for the car to travel 400 m from the start.

10. An elevator car whose floor-to-ceiling distance is equal to 2.7 m starts ascending with constant acceleration 1.2 m/s². 2.0 s after the start a bolt falls from the ceiling of the car. Find :
(a) the bolt’s free fall time
(b) the displacement and the distance covered by the bolt during the free fall in the reference frame fixed to the elevator shaft.
(c) Find displacement bolt with respect to elevator.

11. Two swimmers leave point A on one bank of the river to reach point B lying right across on the other bank. One of them crosses the river along the straight line AB while the other swims at right angles to the stream and then walks the distance that he has been carried away by the stream to get to point B. What was the velocity u of his walking if both swimmers reach the destination simultaneously ? The stream velocity v = 2.0 km/hour and the velocity u of each swimmer with respect to water equals 2.5 km per hour.

12. Two boats, A and B, move away from a buoy anchored at the middle of a river along the mutually perpendicular straight lines ; the boat A along the river, and the boat B across the river. Having moved off an equal distance from the buoy the boats returned. Find the ratio of times of motion of boats τA / τB if the velocity of each boat with respect to water is η = 1.2 times greater than the stream velocity.

13. Point A moves uniformly with velocity v so that the vector v is continually “aimed” at point B which in its turn moves rectilinearly and uniformly with velocity u < v. At the initial moment of time v ⊥ u and the points are separated by a distance l. How soon will the points converge ?

14. The distance between two moving particles P and Q at any time is ‘a’. If vr be their relative velocity and if u and v be the components of vr along and perpendicular to PQ, then show that their closest distance is av/v_r  and that the time that elapses before they arrive at their nearest distance is au/v²_r .

15. An aircraft flies with constant airspeed (speed of aircraft in still air) 200 km/hr from position A to position B, which is 100 km north-east of A and then flies back to A. Throughout the whole flight the wind velocity is 60 km/hr from the west. Find total time of flight from A to B and back.

16. A river of width ‘a’ with straight parallel banks flows due north with speed u. The points O and A are on opposite banks and A is due east of O. Coordinate axes Ox and Oy are taken in the east and north directions respectively. A boat, whose speed is v relative to water, starts from O and crosses the river. If the boat is steered due east and u varies with x as : u = x (a – x) a² v . Find:
(a) equation of trajectory of the boat
(b) time taken to cross the river
(c) absolute velocity of boatman when he reaches the opposite bank.
(d) the displacement of boatman when he reaches the oppsite bank from the initial position.