Motion in Two Dimensions PDF

# Motion in Two Dimensions ## 3.1 Components of Motion - Motion in two dimensions is analyzed by considering the motion of linear components. - The connecting factor between components is time. ### Components of Initial Velocity: - $U_{xo} = v_o cos θ$ - $U_{yo} = v_o sin θ$ ### Components of...

# Motion in Two Dimensions ## 3.1 Components of Motion - Motion in two dimensions is analyzed by considering the motion of linear components. - The connecting factor between components is time. ### Components of Initial Velocity: - $U_{xo} = v_o cos θ$ - $U_{yo} = v_o sin θ$ ### Components of Displacement (constant acceleration only): - $x = x_o + U_{xo}t + \frac{1}{2}a_xt^2 $ - $y= y_o + U_{yo}t + \frac{1}{2}a_yt^2$ - Straight line distance from origin to (x,y): $d = \sqrt{x^2 + y^2}$ magnitude of displacement ### Components of Velocity (constant acceleration only): - $U_x = U_{xo} + a_xt$ - $U_y=U_{yo}+a_yt$ - The magnitude of velocity: $v = \sqrt{v_x^2 + v_y^2}$ combination of the velocities - Direction of velocity: $θ = tan^-1(v_y/v_x)$ relative to the x-axis ## Remark - If velocity constant $a=0$, the motion will continue in a straight line path - If acceleration in the direction of velocity or opposite to it, will continue in a straight line path - If acceleration at some angle other than 0 or 180 to the velocity , the motion will be along a curved path ## 3.2 Vector Addition and Subtraction ### Vector Addition: Geometric Methods #### Triangle Method - Draw first vector (A) from origin. - Draw second vector (B) from tip of first vector. - Draw vector from tail of A to tip of B. This is the the resultant (R). - The vector from the tail of the first vector to the tip of the last vector is the resultant or vector sum R = A + B. - This method is also called the tip-to-tail method. - For more than two vectors, it is called the polygon method. ### Vector Subtraction: - Vector subtraction is a special case of vector addition: - $A - B = A + (-B)$ ## Remark - When two or more vectors are added, they must all have the same units. - 3A a vector with a magnitude three times that of A and pointing in the same direction. - 3A a vector with a magnitude three times that of A and pointing in the opposite direction. - Vector addition is commutative A + B = B + A - Vector addition is associative $(A + B) + Ĉ = A + (B + C)$ - Equality of Two Vectors. Two vectors A and B are equal if they have the same magnitude and the same direction. ## Vector Representation: - Components - Rectangular components - $C_x = C cos⁡ θ$ - $C_y= C sin⁡ θ$ - $C = C_xx + C_yy$ (component form) - Magnitude-angle form. - $C = \sqrt{C_x^2 + C_y^2}$ - $θ = tan^-1(C_y/C_x)$ (magnitude angle form) ## Procedures For Adding Vectors By the Component Method: 1. Resolve the vectors to be added into their x- and y-components. 2. Add all of the x-components together, and all of the y-components together vectorially to obtain the x- and y-components of the resultant, or vector sum. - $C_x = A_x + B_x$ - $C_y = A_y + B_y$ 3. Express the resultant vector, using: - the unit vector component form-for example, C = Cxx + Cyy-or - the magnitude-angle form. - $C = \sqrt{C_x^2 + C_y^2}$ - $θ = tan^-1(C_y/C_x)$ The angle θ is the angle between the resultant and the x-axis in that quadrant.