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Physics · Textbook solutions

Motion in a Plane

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 56 questions

Worked Examples

9 q

Solved Examples

Worked · 9
  1. Eg 3.1
    Rain is falling vertically with a speed of 35 m s−135\,\text{m s}^{-1}. Winds starts blowing after sometime with a speed of 12 m s−112\,\text{m s}^{-1} in east to west direction. In which direction should a boy waiting at a bus stop hold his umbrella?
  2. Eg 3.2
    Find the magnitude and direction of the resultant of two vectors A\mathbf{A} and B\mathbf{B} in terms of their magnitudes and angle θ\theta between them.
  3. Eg 3.3
    A motorboat is racing towards north at 25 km/h25\,\text{km/h} and the water current in that region is 10 km/h10\,\text{km/h} in the direction of 60∘60^{\circ} east of south. Find the resultant velocity of the boat.
  4. Eg 3.4
    The position of a particle is given by r=3.0t i^+2.0t2 j^+5.0 k^\mathbf{r} = 3.0t\,\hat{\mathbf{i}} + 2.0t^{2}\,\hat{\mathbf{j}} + 5.0\,\hat{\mathbf{k}} where tt is in seconds and the coefficients have the proper units for r\mathbf{r} to be in metres. (a) Find v(t)\mathbf{v}(t) and a(t)\mathbf{a}(t) of the particle. (b) Find the magnitude and direction of v(t)\mathbf{v}(t) at t=1.0 st = 1.0\,\text{s}.
  5. Eg 3.5
    A particle starts from origin at t=0t = 0 with a velocity 5.0 i^ m/s5.0\,\hat{\mathbf{i}}\,\text{m/s} and moves in xx-yy plane under action of a force which produces a constant acceleration of (3.0 i^+2.0 j^) m/s2(3.0\,\hat{\mathbf{i}} + 2.0\,\hat{\mathbf{j}})\,\text{m/s}^{2}. (a) What is the yy-coordinate of the particle at the instant its xx-coordinate is 84 m84\,\text{m}? (b) What is the speed of the particle at this time?
  6. Eg 3.6
    Galileo, in his book Two new sciences, stated that "for elevations which exceed or fall short of 45∘45^{\circ} by equal amounts, the ranges are equal". Prove this statement.
  7. Eg 3.7
    A hiker stands on the edge of a cliff 490 m490\,\text{m} above the ground and throws a stone horizontally with an initial speed of 15 m s−115\,\text{m s}^{-1}. Neglecting air resistance, find the time taken by the stone to reach the ground, and the speed with which it hits the ground. (Take g=9.8 m s−2g = 9.8\,\text{m s}^{-2}).
  8. Eg 3.8
    A cricket ball is thrown at a speed of 28 m s−128\,\text{m s}^{-1} in a direction 30∘30^{\circ} above the horizontal. Calculate (a) the maximum height, (b) the time taken by the ball to return to the same level, and (c) the distance from the thrower to the point where the ball returns to the same level.
  9. Eg 3.9
    An insect trapped in a circular groove of radius 12 cm12\,\text{cm} moves along the groove steadily and completes 77 revolutions in 100 s100\,\text{s}. (a) What is the angular speed, and the linear speed of the motion? (b) Is the acceleration vector a constant vector? What is its magnitude?

Exercises

47 q
  1. Ex 3.1
    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.
  2. Ex 3.2
    Pick out the two scalar quantities in the following list : force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.
  3. Ex 3.3
    Pick out the only vector quantity in the following list : Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.
  4. Ex 3.4(a)
    State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (a) adding any two scalars
  5. Ex 3.4(b)
    State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (b) adding a scalar to a vector of the same dimensions
  6. Ex 3.4(c)
    State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (c) multiplying any vector by any scalar
  7. Ex 3.4(d)
    State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (d) multiplying any two scalars
  8. Ex 3.4(e)
    State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (e) adding any two vectors
  9. Ex 3.4(f)
    State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful : (f) adding a component of a vector to the same vector.
  10. Ex 3.5(a)
    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.
  11. Ex 3.5(b)
    Read each statement below carefully and state with reasons, if it is true or false : (b) each component of a vector is always a scalar.
  12. Ex 3.5(c)
    Read each statement below carefully and state with reasons, if it is true or false : (c) the total path length is always equal to the magnitude of the displacement vector of a particle.
  13. Ex 3.5(d)
    Read each statement below carefully and state with reasons, if it is true or false : (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.
  14. Ex 3.5(e)
    Read each statement below carefully and state with reasons, if it is true or false : (e) Three vectors not lying in a plane can never add up to give a null vector.
  15. Ex 3.6(a)
    Establish the following vector inequality geometrically or otherwise : (a) ∣a+b∣≤∣a∣+∣b∣|\mathbf{a}+\mathbf{b}| \le |\mathbf{a}| + |\mathbf{b}| When does the equality sign above apply?
  16. Ex 3.6(b)
    Establish the following vector inequality geometrically or otherwise : (b) ∣a+b∣≥∣ ∣a∣−∣b∣ ∣|\mathbf{a}+\mathbf{b}| \ge \big|\,|\mathbf{a}| - |\mathbf{b}|\,\big| When does the equality sign above apply?
  17. Ex 3.6(c)
    Establish the following vector inequality geometrically or otherwise : (c) ∣a−b∣≤∣a∣+∣b∣|\mathbf{a}-\mathbf{b}| \le |\mathbf{a}| + |\mathbf{b}| When does the equality sign above apply?
  18. Ex 3.6(d)
    Establish the following vector inequality geometrically or otherwise : (d) ∣a−b∣≥∣ ∣a∣−∣b∣ ∣|\mathbf{a}-\mathbf{b}| \ge \big|\,|\mathbf{a}| - |\mathbf{b}|\,\big| When does the equality sign above apply?
  19. Ex 3.7(a)
    Given a+b+c+d=0\mathbf{a} + \mathbf{b} + \mathbf{c} + \mathbf{d} = 0, state whether the following statement is correct : (a) a\mathbf{a}, b\mathbf{b}, c\mathbf{c}, and d\mathbf{d} must each be a null vector.
  20. Ex 3.7(b)
    Given a+b+c+d=0\mathbf{a} + \mathbf{b} + \mathbf{c} + \mathbf{d} = 0, state whether the following statement is correct : (b) The magnitude of (a+c)(\mathbf{a} + \mathbf{c}) equals the magnitude of (b+d)(\mathbf{b} + \mathbf{d}).
  21. Ex 3.7(c)
    Given a+b+c+d=0\mathbf{a} + \mathbf{b} + \mathbf{c} + \mathbf{d} = 0, state whether the following statement is correct : (c) The magnitude of a\mathbf{a} can never be greater than the sum of the magnitudes of b\mathbf{b}, c\mathbf{c}, and d\mathbf{d}.
  22. Ex 3.7(d)
    Given a+b+c+d=0\mathbf{a} + \mathbf{b} + \mathbf{c} + \mathbf{d} = 0, state whether the following statement is correct : (d) b+c\mathbf{b} + \mathbf{c} must lie in the plane of a\mathbf{a} and d\mathbf{d} if a\mathbf{a} and d\mathbf{d} are not collinear, and in the line of a\mathbf{a} and d\mathbf{d}, if they are collinear?
  23. Ex 3.8
    Three girls skating on a circular ice ground of radius 200 m200\,\text{m} start from a point PP on the edge of the ground and reach a point QQ diametrically opposite to PP following different paths as shown in Fig. 3.19. What is the magnitude of the displacement vector for each ? For which girl is this equal to the actual length of path skate ?
  24. Ex 3.9
    A cyclist starts from the centre OO of a circular park of radius 1 km1\,\text{km}, reaches the edge PP of the park, then cycles along the circumference, and returns to the centre along QOQO as shown in Fig. 3.20. If the round trip takes 10 min10\,\text{min}, what is the (a) net displacement, (b) average velocity, and (c) average speed of the cyclist ?
  25. Ex 3.10
    On an open ground, a motorist follows a track that turns to his left by an angle of 60∘60^{\circ} after every 500 m500\,\text{m}. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
  26. Ex 3.11
    A passenger arriving in a new town wishes to go from the station to a hotel located 10 km10\,\text{km} away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km23\,\text{km} long and reaches the hotel in 28 min28\,\text{min}. What is (a) the average speed of the taxi, (b) the magnitude of average velocity ? Are the two equal ?
  27. Ex 3.12
    The ceiling of a long hall is 25 m25\,\text{m} high. What is the maximum horizontal distance that a ball thrown with a speed of 40 m s−140\,\text{m s}^{-1} can go without hitting the ceiling of the hall ?
  28. Ex 3.13
    A cricketer can throw a ball to a maximum horizontal distance of 100 m100\,\text{m}. How much high above the ground can the cricketer throw the same ball ?
  29. Ex 3.14
    A stone tied to the end of a string 80 cm80\,\text{cm} long is whirled in a horizontal circle with a constant speed. If the stone makes 1414 revolutions in 25 s25\,\text{s}, what is the magnitude and direction of acceleration of the stone ?
  30. Ex 3.15
    An aircraft executes a horizontal loop of radius 1.00 km1.00\,\text{km} with a steady speed of 900 km/h900\,\text{km/h}. Compare its centripetal acceleration with the acceleration due to gravity.
  31. Ex 3.16(a)
    Read the statement below carefully and state, with reasons, if it is true or false : (a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre.
  32. Ex 3.16(b)
    Read the statement below carefully and state, with reasons, if it is true or false : (b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point.
  33. Ex 3.16(c)
    Read the statement below carefully and state, with reasons, if it is true or false : (c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector.
  34. Ex 3.17
    The position of a particle is given by r=3.0t i^−2.0t2 j^+4.0 k^ m\mathbf{r} = 3.0t\,\hat{\mathbf{i}} - 2.0t^{2}\,\hat{\mathbf{j}} + 4.0\,\hat{\mathbf{k}}\,\text{m} where tt is in seconds and the coefficients have the proper units for r\mathbf{r} to be in metres. (a) Find the v\mathbf{v} and a\mathbf{a} of the particle? (b) What is the magnitude and direction of velocity of the particle at t=2.0 st = 2.0\,\text{s} ?
  35. Ex 3.18
    A particle starts from the origin at t=0 st = 0\,\text{s} with a velocity of 10.0 j^ m/s10.0\,\hat{\mathbf{j}}\,\text{m/s} and moves in the xx-yy plane with a constant acceleration of (8.0 i^+2.0 j^) m s−2\left(8.0\,\hat{\mathbf{i}} + 2.0\,\hat{\mathbf{j}}\right)\,\text{m s}^{-2}. (a) At what time is the xx-coordinate of the particle 16 m16\,\text{m}? What is the yy-coordinate of the particle at that time? (b) What is the speed of the particle at the time ?
  36. Ex 3.19
    i^\hat{\mathbf{i}} and j^\hat{\mathbf{j}} are unit vectors along xx- and yy-axis respectively. What is the magnitude and direction of the vectors i^+j^\hat{\mathbf{i}} + \hat{\mathbf{j}}, and i^−j^\hat{\mathbf{i}} - \hat{\mathbf{j}} ? What are the components of a vector A=2 i^+3 j^\mathbf{A} = 2\,\hat{\mathbf{i}} + 3\,\hat{\mathbf{j}} along the directions of i^+j^\hat{\mathbf{i}} + \hat{\mathbf{j}} and i^−j^\hat{\mathbf{i}} - \hat{\mathbf{j}}? [You may use graphical method]
  37. Ex 3.20(a)
    For any arbitrary motion in space, state whether the following relation is true : (a) vaverage=(1/2)(v(t1)+v(t2))\mathbf{v}_{\text{average}} = (1/2)\left(\mathbf{v}(t_1) + \mathbf{v}(t_2)\right) (The 'average' stands for average of the quantity over the time interval t1t_1 to t2t_2.)
  38. Ex 3.20(b)
    For any arbitrary motion in space, state whether the following relation is true : (b) vaverage=r(t2)−r(t1)t2−t1\mathbf{v}_{\text{average}} = \dfrac{\mathbf{r}(t_2) - \mathbf{r}(t_1)}{t_2 - t_1} (The 'average' stands for average of the quantity over the time interval t1t_1 to t2t_2.)
  39. Ex 3.20(c)
    For any arbitrary motion in space, state whether the following relation is true : (c) v(t)=v(0)+a t\mathbf{v}(t) = \mathbf{v}(0) + \mathbf{a}\,t (The 'average' stands for average of the quantity over the time interval t1t_1 to t2t_2.)
  40. Ex 3.20(d)
    For any arbitrary motion in space, state whether the following relation is true : (d) r(t)=r(0)+v(0) t+(1/2) a t2\mathbf{r}(t) = \mathbf{r}(0) + \mathbf{v}(0)\,t + (1/2)\,\mathbf{a}\,t^{2} (The 'average' stands for average of the quantity over the time interval t1t_1 to t2t_2.)
  41. Ex 3.20(e)
    For any arbitrary motion in space, state whether the following relation is true : (e) aaverage=v(t2)−v(t1)t2−t1\mathbf{a}_{\text{average}} = \dfrac{\mathbf{v}(t_2) - \mathbf{v}(t_1)}{t_2 - t_1} (The 'average' stands for average of the quantity over the time interval t1t_1 to t2t_2.)
  42. Ex 3.21(a)
    Read the 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
  43. Ex 3.21(b)
    Read the statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (b) can never take negative values
  44. Ex 3.21(c)
    Read the statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (c) must be dimensionless
  45. Ex 3.21(d)
    Read the statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (d) does not vary from one point to another in space
  46. Ex 3.21(e)
    Read the statement below carefully and state, with reasons and examples, if it is true or false : A scalar quantity is one that (e) has the same value for observers with different orientations of axes.
  47. Ex 3.22
    An aircraft is flying at a height of 3400 m3400\,\text{m} above the ground. If the angle subtended at a ground observation point by the aircraft positions 10.0 s10.0\,\text{s} apart is 30∘30^{\circ}, what is the speed of the aircraft ?