MOTION IN A PLANE

dimensions. If α , β , and γ are the angles * between A and the x -, y -, and z -axes, respectively [Fig. . (d)], we have (d) A cos , A A cos , A A cos α β γ ( .16a) In general, we have ( .16b) The magnitude of vector A is ( .16c) A position vector r can be expressed as ( .

) where x, y , and z are the components of r along x-, y-, z- axes, respectively. . VECTOR ADDITION – ANALYTICAL METHOD Although the graphical method of adding vectors helps us in visualising the vectors and the resultant vector, it is sometimes tedious and has limited accuracy. It is much easier to add vectors by combining their respective components.

Consider two vectors A and B in x - y plane with components A x , A y and B x , B y : ( . ) Fig. . Answer Let OP and O Q represent the two vectors A and B making an angle θ (Fig.

. ). Then, using the parallelogram method of vector addition, OS represents the resultant vector R : R = A + B SN is normal to OP and PM is normal to OS . From the geometry of the figure, OS = ON + SN but ON = OP + PN = A + B cos θ SN = B sin θ OS = ( A + B cos θ ) + ( B sin θ ) or, R = A + B + AB cos θ 2AB cos θ ( .24a) In ∆ OSN, SN = OS sin α = R sin α , and in ∆ PSN, SN = PS sin θ = B sin θ Therefore, R sin α = B sin θ or, α ( .24b) Similarly, PM = A sin α = B sin β or, β α ( .24c) Combining Eqs.

( .24b) and ( .24c), we get β α ( .24d) Using Eq. ( .24d), we get: α ( .24e) where R is given by Eq. ( .24a). or, cos SN OP PN α ( .24f) Equation ( .24a) gives the magnitude of the resultant and Eqs.

( .24e) and ( .24f) its direction. Equation ( .24a) is known as the Law of cosines and Eq. ( .24d) as the Law of sines . ⊳ Example .

A motorboat is racing towards north at km/h and the water current in that region is km/h in the direction of ° east of south. Find the resultant velocity of the boat. Answer The vector v b representing the velocity of the motorboat and the vector v c representing the water current are shown in Fig. .

in directions specified by the problem. Using the parallelogram method of addition, the resultant R is obtained in the direction shown in the figure. Fig. .

We can obtain the magnitude of R using the Law of cosine : v v cos120 - / km/h ) ≅ To obtain the direction, we apply the Law of sines v c φ or, sin φ c sin sin120 . . . ≅ φ ≅ .

. MOTION IN A PLANE In this section we shall see how to describe motion in two dimensions using vectors. . .

Position Vector and Displacement The position vector r of a particle P located in a plane with reference to the origin of an x-y reference frame (Fig. . ) is given by where x and y are components of r along x -, and y - axes or simply they are the coordinates of the object. (a) (b) Fig.

. (a) Position vector r . (b) Displacement ∆ r and average velocity v of a particle. Suppose a particle moves along the curve shown by the thick line and is at P at time t and P ′ at time t ′ [Fig.

. (b)]. Then, the displacement is : ∆ r = r ′ – r ( . ) and is directed from P to P ′ .

We can write Eq. ( . ) in a component form: