Kinematics Exercise-1

KINEMATICS Page # 15 Exercise - I UNSOLVED PROBLEMS Q.1 In which of the following examples of motion, can the body be considered approximately a point object : (a) a railway carriage moving without jerks between two stations. (b) a monkey sitting on top of a man cycling smoothly on a circular track. (c) a spinning cricket ball that turns sharply on hitting the ground. (D) a tumbling beaker that has slipped off the edge of a table. Q.2 The position-time (x-t) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in figure. Choose the correct entries in the brackets below : x Q P A B O t (a) (A/B) lives closer to the school than (B/A) (b) (A/B) starts from the school earlier than (B/A) (c) (A/B) walks faster than (B/A). (d) A and B reach home at the (same/different) time (e) (A/B) overtakes (B/A) on the road (once/twice). Q.3 A woman starts from her home at 9.00 am, walks with a speed of 5 km h–1 on a straight road up to her office 2.5 km away, stays at the office up to 5.00 pm, and returns home by an auto with a speed of 25 km h–1. Choose suitable scales and plot the x-t graph of her motion. Q.4 A drunkard walking in a narrow lane takes 5 steps forward and 3 steps backward, followed again by 5 steps forward and 3 steps backward, and son on. Each step is 1 m long and requires 1 s Plot the x-t graph of his motion. Determine graphically and otherwise how long the drunkard takes to fall in a pit 13 m away from the start. Q.5 A jet airplane travelling at the speed of 500 km h –1 ejects its products of combustion at the speed of 1500 km h–1 relative to the jet plane. What is the speed of the latter with respect to an observer on the ground ? Q.6 A car moving along a straight highway with speed of 126 km h–1 is brought to a stop within a distance of 200 m. What is the retardation of the car (assumed uniform), and how long does it take for the car to stop ? Q.7 Two trains A and B of length 400 m each are moving on two parallel tracks with a uniform speed of 72 km h–1 in the same direction, with A ahead of B. The driver of B decides to overtake A and accelerates by 1 ms–2. If after 50 s, the guard of B just brushes past the driver of A, what was the original distance between them (guard of B and driver of A) ? Q.8 On a two-lane road, car A is travelling with a constant speed of 36 km h–1. Two cars B and C approach car A in opposite directions with a constant speed of 54 km h –1 each. At a certain instant, when the distance AB is equal to AC, both being 1 km, B decides to overtake A before C does. What minimum acceleration of car B is required to avoid an accident Q.9 Two towns A and B are connected by a regular bus service with a bus leaving in either direction every T minutes. A man cycling with a speed of 20 km h–1 in the direction A to B notices that a bus goes past him every 18 min in the direction of his motion, and every 6 min in the opposite direction. What is the period T of the bus service and with what speed (assumed constant) do the buses ply on the road ? Q.10 A player throws a ball upwards with an initial speed of 29.4 m s–1. (a) What is the direction of acceleration during the upward motion of the ball ? (b) What are the velocity and acceleration of the ball at the highest point of its motion ? (c) Choose the x = 0 m and t = 0 s to be the location and time of the ball at its highest point, vertically downward direction to be the positive direction of x-axis, and give the signs of position, velocity and acceleration of the ball during its upward, and downward motion. (d) To what height does the ball rise and after how long does the ball return to the player's hand ? (Take g = 9.8 m s–2 and neglect air resistance). Q.11 Read each statement below carefully and state with reasons and examples (if possible) if it is true or false; A particle in one-dimensional motion (a) with zero speed at an instant may have nonzero acceleration at that instant . (b) with zero speed may have non-zero velocity. (c) with constant speed must have zero acceleration (d) with positive value of acceleration must be speeding up. KINEMATICS Page # 16 Q.12 A ball is dropped from a height of 90 m on a floor. At each collision with the floor, the ball loses one tenth of its speed. Plot the speed-time graph of its motion between t = 0 to 12 s. Q.13 Explain clearly, with examples, the distinction between : (a) magnitude of displacement (sometimes called distance) over an interval of time, and the total length of path covered by a particle over the same interval : (b) magnitude of average velocity over an interval of time and the average speed over the same interval. [Average speed of a particle over an interval of time is defined as the total path length divided by the time interval.] Show in both (a) and (b) that the second quantity is either greater than or equal to the first. When is the equality sign true ? [For simplicity, consider one-dimensional motion only]. Q.14 A man walks on a straight road from his home to a market 2.5 km away with a speed of 5 km h–1. finding the market closed, he instantly turns and walks back home with a speed of 7.5 km h–1 . What is the (a) magnitude of average velocity, and (b) average speed of the man over the interval of time (i) 0 to 30 min, (ii) o to 50 min, (iii) 0 to 40 min ? [Note : You will appreciate from this exercise why it is better to define average speed as total path length divided by time, and not as magnitude of average velocity. You would not like to tell the tired man on his return home that his average speed was zero !]. Q.35 i n 3 . 1 3 a n d 3 . 1 4 , w e h a v e c a r e f u l l y d i s t i n g u i s h e d b e t w ee n a v e r a g e s p e e d a n d magnitude of average velocity. No such distinction necessary when we consider instantaneous speed and magnitude of velocity. The instantaneous speed is always equal to the magnitude of instantaneous velocity. Why ? Q.16 Look at the graphs (a) to (d) Fig. 3.20) carefully and state, with reasons , which of these cannot possibly represent one-dimensional motion of a particle. (c) (d) t t Q.17 Figures 3.21 shows the x-t plot of onedimensional motion of a particle. Is it correct to say from the graph that the particle moves in a straight line for t < 0 and on a parabolic path for t > 0 ? If not, suggest a suitable physical context for this graph. x 0 t Q.18 A police van moving on a highway with a speed of 30 km h–1 fires a bullet at a thief's car speeding away in the same direction with a speed of 192 km h–1 . If the muzzle speed of the bullet is 150 m s–1, with what speed does the bullet hit the thief's car ? (Note : Obtain that speed which is relevant for damaging the thief's car). Q.19 Suggest a suitable physical situation for each of the following graphs (fig. 3.22)_ : x v v A t B t t Q.20 Figure 3.23 gives the x-t plot of a particle executing one-dimensional simple harmonic motion. (You will learn about this motion in more detail in Chapter 14). Give the signs of position, velocity and acceleration variables of the particle at t = 0.3 s, 1.2 s, –1.2 s. x -2 -1 0 1 2 3 t v x (a) Total path length speed t (b) t Q.21 Figure 3.24 gives the x-t plot of a particle in one-dimensional motion. Three different equal intervals of time are shown. In which interval is the average speed greatest, and in which is it the least ? Give the sign of average velocity for each interval. KINEMATICS Page # 17 x t D speed B C A 1 2 3 t Q.23 A three-wheeler starts from rest, accelerates uniformly with 1 m s–2 on a straight road for 10 s, and then moves with uniform velocity. Plot the distance covered by the vehicle during the nth second (n = 1, 2, 3, ....) versus n. What do you expect this plot to be during accelerated motion : a straight line or a parabola ? Q.24 A boy standing on a stationary lift (open from above) throws a ball upwards with the maximum initial speed he can, equal to 49 ms–1. How much time does the ball take to return to his hands ? If the lift starts moving up with a uniform speed of 5 ms–1 and the boy again throws the ball up with the maximum speed he can, how long does the ball take to return to his hands ? If the lift starts moving up with a uniform accleration of 5 ms–2 and the boy again throws the ball up with the maximum speed he can, how long does the ball take to return to his hands ? (g = 9.8 m/s2) Q.25 On a long horizontally moving belt (figure 3.26), a child runs to and fro with a speed 9 km h–1 (with respect to the belt) between his father and mother located 50 m apart on the moving belt. The belt moves with a speed of 4 km h–1. For an observer on a stationary platform outside, what is the (a) speed of the child running in the direction of motion of the belt ? (b) speed of the child running opposite to the direction of motion of the belt ? (c) time taken by the child in (a) and (b) ? Which of the answers alter if motion is viewed by one of the parents ? Q.26 Two stones are thrown up simultaneously from the edge of a cliff 200 m high with initial speeds of 15 m s–1 and 30 m s–1. Verify that the graph shown in figure 3.27 correctly represents the time variation of the relative position of the second stone with respect to the first. Neglect air resistance and assume that the stones do not rebound after hitting the ground. Take g = 10 m s– 2 . give the equations for the linear and curved parts of the plot. 120 100 80 60 40 20 2 4 6 8 t(s) 10 Q.27 The speed-time graph of a particle moving along a fixed direction is shown in figure 3.28. Obtain the distance traversed by the particle between (a) t = 0 s to 10 s. –1 Q.22 Figure 3.25 gives a speed-time graph of a particle in motion along a constant direction. Three equal intervals of time are shown. In which interval is the average acceleration greatest in magnitude ? In which interval is the average speed greatest " Choosing the positive direction as the constant direction of motion, give the signs of v and a in the three intervals. What are the accelerations at the points A, B, C and D ? speed (ms ) 2 x2 -x1(m) 3 1 12 0 5 10 t What is the average speed of the particle over the intervals in (a) and (b) ? Q.28 The velocity-time graph of a particle in onedimensional motion is shown in figure. v 0 t1 t2 t KINEMATICS Which of the following formulae are correct for describing the motion of the particle over the timeinterval t1 to t2 : (a) x(t2) = x(t1) + v (t1) (t2 – t1) + (½) a (t2 – t1)2 (b) x(t2) = v (t1) + a (t2 – t1) (c) v arverage= (x (t2) – x (t1))/ (t2 – t1) (d) aarverage= (v (t2) – v (t1))/ (t2 – t1) (e) x(t2) =x (t1) + v arverage(t2–t1) +(½) aaverage(t2–t1)2 (f) x (t2) – x (t1) = area under the v-t curve bounded by the t-axis and the dotted line shown. Q.29 State, for each of the following physical quantities, if it is a scalar or a vector : volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity. Q.30 Pick out the two scalar quantities in the following list : force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, reaction as per Newton's third law, relative velocity. Q.31 Pick out the only vector quantity in the following list : Temperature, pressure, impulse, time, power , total path lengthy, energy, gravitational potential, coefficient of friction, charge. Q.32 State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (a) adding any two scalars, (b) adding a scalar to a vector of the same dimensions, (c) multiplying any vector by any scalar, (d) multiplying any two scalars, (e) adding any two vectors, (f) adding a component of a vector to the same vector. Q.33 Read each statement below carefully and state with reasons, if it is true or false : (a) The magnitude of a vector is always a scalar, (b) each component of a vector is always a scalar, (c) the total path length is always equal to the magnitude of the displacement vector of a particle. (d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time, (e) Three vectors not lying in a plane can never add up to give a null vector. Q.34 Establish the following vector inequalities geometrically or otherwise : (a) |a + b| < |a| + |b| (b) |a + b| > |a| – |b| (c) |a – b| < |a| + |b| (d) |a – b| > |a| – |b| Page # 18 When does the equality sign above apply ? Q.35 Given a + b + c + d = 0, which of the following statements are correct : (a) a, b, c and d must each be a null vector, (b) The magnitude of (a + c) equals the magnitude of (b + d), (c) The magnitude of a can never be greater than the sum of the magnitudes of b, c and d, (d) b + c must lie in the plane of a and d if a and d are not collinear, and in the line of a and d, if they are collinear ? Q.36 The girls skating on a circular ice ground of radius 200 m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in figure 4.23. What is the magnitude of the displacement vector for each ? For which girls is this equal to the actual length of path skate ? Q A B P Q.37 A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along OQ as shown in figure 4.24. If the round trip takes 10 min, what is the (a) net displacement, (b) average velocity, and (c) average speed of the cyclist ? Q O P Q.38 On an open ground, a motorist follows a track that turns to his left by an angle of 60° after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eight turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case. Q.39 A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (a) the average speed of the taxi, (b) the magnitude of average velocity ? Are the two equal ? KINEMATICS Page # 19 Q.40 Rain is falling vertically with a sped of 30 m s–1. A woman rides a bicycle with a speed of 10 m s–1 in the north to south direction. What is the direction in which she should hold her umbrella ? Q.41 A man can swim with a sped of 4 km/h in still water. How long does he take to cross a river 1 km wide if the river flows steadily at 3 km/h and he makes his strokes normal to the river current ? How far down the river does he go when he reaches the other bank Q.42 In a harbour, wind is blowing at he speed of 72 km/h and the flag on the mast of a boat anchored in the harbour flutters along the N-E direction. If the boat starts moving at a sped of 51 km/h to the north, what is the direction of the flag on the mast of the boat ? Q.43 The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that a ball thrown with a speed of 40 m s–1 can go without hitting the ceiling of the hall ? Q.44 A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much high above the ground can the cricketer throw the same ball ? Q.45 A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and direction of acceleration of the stone where t is in seconds and the coefficients have the proper units for r to be in metres. (a) Find the v and a of the particle ? (b) What is the magnitude and direction of velocity of the particle at t=2s? Q.49 A particle starts from the origin at t = 0 s with a velocity of 10.0 ĵ m/s and moves in the x-y plane with a constant acceleration of ( 8.0 î  2.0 ĵ ) m s–2. (a) At what time is the x-coordinate of the particle 16 m ? What is the y-coordinate of the particle at that time ? (b) What is the speed of the particle at the time ? Q.50 î and ĵ are unit vector along x- and y-axis respectively. What is the magnitude and direction of the vectors î  ĵ , and î  ĵ ? What are the components of a vector A = 2 î  3 ĵ along the directions of î  ĵ and î  ĵ ? Q.51 For any arbitrary motion in space, which of the following relations are true : (a) Vaverage = (1/2) (v (t1) + v(t2)) (b) Vaverage = [ r (t2) – r (t1)] / (t2 – t1) (c) V (t) = v (0) at (d) r(t) = r(0) + v(0) t + (1/2) a t2 (e) aaverage = [v (t2) – v (t1)] / (t2 – t1) Q.46 An aircraft executes a horizontal loop of radius 1 km with a steady speed of 900km/h. Compare its centripetal acceleration with the acceleration due to gravity. (The 'average' stands for average of the quantity over Q.47 Read each statement below carefully and state, with reasons, if it is rue or false : (a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre (b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point (c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector. coordinate axes ? Q.48 The position of a particle is given by r r  3.0 t î  2.0 t ĵ  4.0 k̂ m 2 the time interval t1 to t2 ) Q.52 Which of the following quantities are independent of the choice of orientation of the a + b, 3ax + 2by, |a + b – c|, angle between b and c, a where is a scalar ? Q.53 Read each statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (a) is conserved in a process (b) can never take negative values (c) must be dimensionless (d) does not vary from one point to another in space (e) has the same value for observers with different orientations of axes.