Study Material
MOTION IN A PLANE
Very Short Answer Questions
Q1. Vector \(3\mathbf{i}+4\mathbf{j}\). Find the magnitude of the vector and the angle it makes with the x-axis.
Q2. Define scalar and vector quantities with examples.
Q3. What is a unit vector? Write the unit vectors along the x-, y-, and z-axes.
Q4. State the conditions under which two vectors are said to be equal.
Q5. What is the physical significance of a null (zero) vector?
Q6. State the triangle law of vector addition.
Q7. What is meant by the resolution of a vector?
Q8. Write the expressions for the x- and y-components of a vector of magnitude \(A\) inclined at an angle \(\theta\) with the x-axis.
Q9. What is centripetal acceleration? Write its expression in terms of angular velocity and radius.
Q10. State the relation between linear velocity and angular velocity in uniform circular motion.
Q11. Write the equation for the trajectory (path) of a projectile.
Q12. Rain drops are falling vertically at \(35\,\mathrm{m\,s^{-1}}\) and a wind is blowing at \(12\,\mathrm{m\,s^{-1}}\) from east to west. In which direction should a person hold an umbrella, so that the rain drops do not fall on him?
Q13. A projectile is launched with a speed of \(20\,\mathrm{m\,s^{-1}}\) at an angle of \(30^\circ\) with the horizontal. Calculate the horizontal component of its velocity.
Q14. Two vectors of magnitudes \(5\) units and \(12\) units are perpendicular to each other. Find the magnitude of their resultant vector.
Q15. A particle's position vector is given by \(\mathbf{r}=4t\mathbf{i}+t^2\mathbf{j}\). Find its velocity vector at \(t=3\,\mathrm{s}\).
Q16. A particle moves with a constant acceleration \(\mathbf{a}=3\mathbf{i}+2\mathbf{j}\,\mathrm{m\,s^{-2}}\). If initial velocity is \(4\mathbf{i}\,\mathrm{m\,s^{-1}}\), find its velocity after \(2\) seconds.
Q17. A cricket ball is hit with an initial velocity of \(28\,\mathrm{m\,s^{-1}}\) at \(30^\circ\) above the horizontal. What is the time taken by the ball to reach the highest point? (Take \(g=9.8\,\mathrm{m\,s^{-2}}\))
Q18. An object is projected such that it reaches a maximum height of \(20\,\mathrm{m}\). What was its vertical component of initial velocity? (Take \(g=9.8\,\mathrm{m\,s^{-2}}\))
Q19. Find the angular speed of an object completing 5 revolutions in 2 seconds.
Q20. An object is in uniform circular motion with a speed of \(10\,\mathrm{m\,s^{-1}}\) along a circular path of radius \(5\,\mathrm{m}\). Calculate its centripetal acceleration.
Q1. Explain the analytical method of vector addition and derive expressions for the components of the resultant vector.
Q2. State parallelogram law of vectors. Obtain an expression for the magnitude of the resultant of two vectors.
Q3. Derive the equations of motion in a plane with constant acceleration using vector notation.
Q4. Derive expressions for the time of flight and maximum height of a projectile.
Q5. What is uniform circular motion? Derive an expression for centripetal acceleration in terms of angular speed.
Q6. Describe how velocity and acceleration vectors are represented graphically in curved motion.
Q7. Derive an expression for the horizontal range of a projectile. Show that the range is maximum, when the angle of projection is \(45^\circ\).
Q8. Write the vector form of the position, velocity, and acceleration of a particle in a plane and explain each term.
Q9. Show that the path followed by projectile is a parabola.
Q10. Define scalar product. Write its properties.
Q11. Define vector product. Write its properties.
Q12. If \(|\mathbf{A}+\mathbf{B}|=|\mathbf{A}-\mathbf{B}|\), then find the angle between \(\mathbf{A}\) and \(\mathbf{B}\).
Q1. Ship A is \(10\,\mathrm{km}\) due west of ship B. Ship A is heading directly north at a speed of \(30\,\mathrm{km\,h^{-1}}\), while ship B is heading in a direction \(60^\circ\) west of north at a speed of \(20\,\mathrm{km\,h^{-1}}\). (i) Determine the magnitude of the velocity of ship B relative to ship A. (ii) What will be their distance of closest approach?
Q2. If \(\theta\) is the angle of projection, \(R\) the range, \(h\) the maximum height, \(T\) the time of flight then show that (a) \(\tan\theta=4h/R\) and (b) \(h=gT^2/8\).
Q3. A projectile is fired at an angle of \(60^\circ\) to the horizontal with an initial velocity of \(800\,\mathrm{m\,s^{-1}}\) from ground: (i) Find the time of flight of the projectile before it hits the ground. (ii) Find the horizontal distance it travels before it hits the ground (range). (iii) Find the time taken by the projectile to reach its maximum height.
Q4. For a particle projected slantwise from the ground, the magnitude of its position vector with respect to the point of projection, when it is at the highest point of the path is found to be \(\sqrt{2}\) times the maximum height reached by it. Show that the angle of projection is \(\tan^{-1}(2)\).
Q5. An object is launched from a cliff \(20\,\mathrm{m}\) above the ground at an angle of \(30^\circ\) above the horizontal with an initial speed of \(30\,\mathrm{m\,s^{-1}}\). How far horizontally does the object travel before landing on the ground? (\(g=10\,\mathrm{m\,s^{-2}}\))
Q6. O is a point on the ground chosen as origin. A body first suffers a displacement of \(10\sqrt{2}\,\mathrm{m}\) North-East, next \(10\,\mathrm{m}\) North and finally \(10\sqrt{2}\,\mathrm{m}\) North-West. How far it is from the origin?
Q7. From a point on the ground, a particle is projected with an initial velocity \(u\), such that its horizontal range is maximum. Find the magnitude of average velocity during its ascent.
Q8. A particle is projected from the ground with some initial velocity making an angle of \(45^\circ\) with the horizontal. It reaches a height of \(7.5\,\mathrm{m}\) above the ground while it travels a horizontal distance of \(10\,\mathrm{m}\) from the point of projection. Find the initial speed of projection. (\(g=10\,\mathrm{m\,s^{-2}}\))
Q9. Wind is blowing from the south at \(5\,\mathrm{m\,s^{-1}}\). To a cyclist it appears to be blowing from the east at \(5\,\mathrm{m\,s^{-1}}\). Find the velocity of the cyclist.
Q10. A person walking at \(4\,\mathrm{m\,s^{-1}}\) finds rain drops falling slantwise on to his face with a speed of \(4\,\mathrm{m\,s^{-1}}\) at an angle of \(30^\circ\) with the vertical. Show that the actual speed of the rain drops is \(4\,\mathrm{m\,s^{-1}}\).
Q11. A ball is tossed from the window of a building with an initial velocity of \(8\,\mathrm{m\,s^{-1}}\) at an angle of \(20^\circ\) below the horizontal. It strikes the ground \(3\,\mathrm{s}\) later. From what height was the ball thrown? How far from the base of the building does the ball strike the ground?
Q12. Two balls are projected from the same point in directions \(30^\circ\) and \(60^\circ\) with respect to the horizontal. What is the ratio of their initial velocities if they (a) attain the same height? (b) have the same range?
Q1. Rain is falling vertically with a speed of \(35\,\mathrm{m\,s^{-1}}\). Winds starts blowing after sometime with a speed of \(12\,\mathrm{m\,s^{-1}}\) in east to west direction. In which direction should a boy waiting at a bus stop hold his umbrella?
Q2. Find the angle between force \(\mathbf{F}=(3\mathbf{i}+4\mathbf{j}-5\mathbf{k})\) unit and displacement \(\mathbf{d}=(5\mathbf{i}+4\mathbf{j}+3\mathbf{k})\) unit. Also find the projection of \(\mathbf{F}\) on \(\mathbf{d}\).
Q3. Find the scalar and vector products of two vectors \(\mathbf{a}=(3\mathbf{i}-4\mathbf{j}+5\mathbf{k})\) and \(\mathbf{b}=(-2\mathbf{i}-\mathbf{j}-3\mathbf{k})\).
Q4. The position of a particle is given by \(\mathbf{r}=3.0t\mathbf{i}+2.0t^2\mathbf{j}+5\mathbf{k}\), where \(t\) is in seconds and the coefficients have the proper units for \(\mathbf{r}\) to be in metres. (a) Find \(\mathbf{v}(t)\) and \(\mathbf{a}(t)\) of the particle. (b) Find the magnitude and direction of \(\mathbf{v}(t)\) at \(t=1.0\,\mathrm{s}\).
Q5. A particle starts from origin at \(t=0\) with a velocity \(5.0\mathbf{i}\,\mathrm{m\,s^{-1}}\) and moves in x-y plane under action of a force which produces a constant acceleration of \((3.0\mathbf{i}+2.0\mathbf{j})\,\mathrm{m\,s^{-2}}\). Find (a) the y-coordinate of the particle at the instant its x-coordinate is \(84\,\mathrm{m}\), and (b) the speed of the particle at this time.
Q6. Rain drops are falling vertically with a speed of \(35\,\mathrm{m\,s^{-1}}\). A woman rides a bicycle with a speed of \(12\,\mathrm{m\,s^{-1}}\) in east to west direction. What is the direction in which she should hold her umbrella so that rain drops will not fall on her.
Q7. Galileo, in his book Two new sciences, stated that “for elevations which exceed or fall short of \(45^\circ\) by equal amounts, the ranges are equal”. Prove this statement.
Q8. A hiker stands on the edge of a cliff \(490\,\mathrm{m}\) above the ground and throws a stone horizontally with an initial speed of \(15\,\mathrm{m\,s^{-1}}\). Neglecting air resistance, find the time taken by the stone to reach the ground, and the speed with which it hits the ground. (Take \(g=9.8\,\mathrm{m\,s^{-2}}\)).
Q9. A cricket ball is thrown at a speed of \(28\,\mathrm{m\,s^{-1}}\) in a direction \(30^\circ\) 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.
Q10. An insect trapped in a circular groove of radius \(12\,\mathrm{cm}\) moves steadily along the groove and completes 7 revolutions in \(100\,\mathrm{s}\). (a) What is the angular speed, and the linear speed of the insect? (b) Does the acceleration vector remain constant? What is its magnitude?
Q11. A particle moves along a circular path of radius \(10\,\mathrm{m}\). Its linear speed is given by \(v=3t\), where \(t\) is in second and \(v\) is in \(\mathrm{m\,s^{-1}}\). (a) Find the centripetal and tangential acceleration at \(t=2\,\mathrm{s}\). (b) Calculate the angle between resultant acceleration and the radius vector.