Chapter 3 ยท Introduction

Chapter 3 โ€” Motion in a Plane

Motion in two dimensions needs quantities with direction. This chapter builds the language of vectors and uses it to understand thrown balls, water jets and anything that goes round in a circle.

The ball is hit towards the boundary. A fielder sprints, gathers it $70\ \text{m}$ from the stumps and throws it on the full to the wicket-keeper. In the two or three seconds of its flight the ball does two quite different things at once: it moves sideways across the field at an almost steady speed, and it rises and falls exactly like a stone thrown straight up. A good fielder never thinks about this, but his arm has learned the answer to a precise question โ€” at what speed and angle must the ball leave the hand to arrive at the keeper's gloves, and how long will it take?

To ask such questions we need a new kind of quantity. In Chapter 2 a plus or minus sign was enough to show direction along a line; in a plane there are infinitely many directions. Vectors carry a magnitude and a direction, and they add by a triangle rule rather than by plain arithmetic. With vectors, and the trick of splitting them into perpendicular components, motion in a plane becomes two straight-line motions running side by side. That one idea explains the parabola of every projectile, why a $45^\circ$ throw goes furthest, and โ€” with one more step โ€” why anything moving round a circle at constant speed is accelerating towards the centre, from a ceiling fan to the Moon.

Scalars & vectorsโ†’Triangle lawโ†’Componentsโ†’Velocity & acceleration in a planeโ†’Constant accelerationโ†’Projectilesโ†’Circular motionโ†’$a_c=v^2/r$
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • tell scalars from vectors, multiply a vector by a number, and add and subtract vectors by the triangle and parallelogram laws;
  • resolve a vector into components, use unit vectors $\hat{\mathbf{i}}$, $\hat{\mathbf{j}}$, $\hat{\mathbf{k}}$, add vectors analytically, and use $R^2=A^2+B^2+2AB\cos\theta$;
  • describe motion in a plane with position, velocity and acceleration vectors, and solve problems with constant acceleration by treating $x$ and $y$ separately;
  • find the trajectory, time of flight, maximum height and range of a projectile, and explain why complementary angles give equal ranges;
  • relate angular speed, period, frequency and speed in uniform circular motion, and derive and use the centripetal acceleration $a_c=v^2/r=\omega^2r$;
  • solve Board, JEE Main, JEE Advanced and NEET-style problems on motion in a plane (including relative velocity in two dimensions, as an exam extension).
Teaser โ€” The Fountain of Five Jets

A fountain nozzle sends out five jets with the same speed at $15^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $75^\circ$ above the horizontal (air resistance neglected). Before you press Play, predict: which jet lands furthest away, and do any two land at the same spot? Then change the speed and see whether the pattern changes.

12 m/s
range at $45^\circ$โ€”
$30^\circ$ and $60^\circ$โ€”
$15^\circ$ and $75^\circ$โ€”

Sections in this chapter

3.1

Scalars, Vectors and Graphical Vector Addition

Quantities with direction, position and displacement vectors, multiplying by numbers, the triangle and parallelogram laws, and subtraction.

$|A-B|\le|\mathbf{A}+\mathbf{B}|\le A+B$
Open section โ†’3.2

Resolution of Vectors and Analytical Addition

Unit vectors, components, adding vectors by components, and the resultant of two vectors at any angle.

$R^2=A^2+B^2+2AB\cos\theta$
Open section โ†’3.3

Motion in a Plane with Constant Acceleration and Projectile Motion

Velocity and acceleration vectors, independent $x$ and $y$ motions, projectiles and their range.

$R=\dfrac{u^2\sin2\theta}{g}$
Open section โ†’3.4

Uniform Circular Motion

Angular speed, period and frequency, and the centripetal acceleration of a body moving round a circle.

$a_c=\dfrac{v^2}{r}=\omega^2r$
Open section โ†’
โœ… Before you start
From Chapter 2 you need displacement and velocity (Section 2.1), acceleration and motion graphs (Section 2.2), and the kinematic equations with free fall (Section 2.3) โ€” Chapter 3 uses them along each axis separately. From Class 10 mathematics you need Pythagoras' theorem, the trigonometric ratios $\sin$, $\cos$, $\tan$ and their values at $30^\circ$, $45^\circ$ and $60^\circ$, and solving quadratic equations. From Chapter 1, recall the radian (Section 1.1) and significant figures (Section 1.2). Rates of change are used exactly as in Chapter 2.
โ† 2.3 Kinematic Equations and Free Fall3.1 Scalars, Vectors and Graphical Vector Addition โ†’