motion in two Dimensions
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Motion in Two Dimensions
Topic Overview: This chapter discusses how objects move in a two-dimensional plane, utilizing vector algebra.
Key Topics Covered:
Reminder of vectors and vector algebra
Displacement and position in 2-D
Average and instantaneous velocity in 2-D
Average and instantaneous acceleration in 2-D
Projectile motion
Uniform circular motion
Relative velocity
Vector Addition and Subtraction
Analytical (Trigonometry) Method:
When vectors form a right triangle with each other:
Use the Pythagorean theorem to find the magnitude of the resultant vector.
Use the inverse trigonometric function to find the angle of the resultant vector.
Notations:
Let ( A ) = magnitude of the resultant vector.
( heta ) = angle,
Trigonometric Relationships:
SOH – CAH – TOA:
( H = \text{hypotenuse of triangle} )
( \sin \theta = \frac{O}{H} )
( \cos \theta = \frac{A}{H} )
( \tan \theta = \frac{O}{A} = \frac{O}{H} )
Inverse functions:
( \theta = \sin^{-1}(O) )
( \theta = \cos^{-1}(A) )
( \theta = \tan^{-1}\left(\frac{O}{H}\right) )
Vector and Its Components
The components of the vector are the legs of the right triangle whose hypotenuse is ( A ):
( \
ho A = Ax + Ay ) where:( A_x = A \cdot \cos(\theta) )
( A_y = A \cdot \sin(\theta) )
Kinematic Variables:
In One Dimension:
Position: ( x(t) ) in meters (m)
Velocity: ( v(t) ) in meters per second (m/s)
Acceleration: ( a(t) ) in meters per second squared (m/s²)
In Three Dimensions:
Position: ( x, y, z )
Velocity: ( vx, vy, v_z )
Acceleration: ( ax, ay, a_z )
Motion Equations in Two Dimensions
Independence of Components: Each dimension can have independent motions.
Constant Acceleration Equations:
Used for problem-solving where:
Time ( t = 0 ) is the beginning of motion;
Accelerations ( ax ) and ( ay ) are constants.
Equations:
Initial velocity ( v_0 ) and initial displacement integrated into:
( \mathbf{v} = \mathbf{v_0} + \mathbf{a} t )
( \mathbf{r} = \mathbf{r0} + \mathbf{v0} t + \frac{1}{2}\mathbf{a} t^2 )
Problem Solving Steps
For solving questions involving motion in two dimensions:
Define the coordinate system with axes and origin.
List known quantities including initial velocities, accelerations, and initial conditions.
Identify equations of motion relevant to the problem and make sure all parts of motion are considered (both x and y components).
Ensure the time ( t ) is consistent when analyzing x and y components.
Example Problem: Motion of a Turtle
A turtle starts at the origin and moves with a speed ( v_0 = 10 \, cm/s ) at an angle of ( 25° ) to the horizontal.
Questions:
(a) Find the coordinates of the turtle after 10 seconds.
(b) Determine the total distance the turtle traveled in that time.
Method of Solution:
Components must be resolved independently for horizontal (x) and vertical (y) motions:
For x component:
[ v{x 0} = v0 \cdot \cos(25°) = 10 \cdot \cos(25°) = 9.06 \, cm/s ]
[ \Delta x = v_x \cdot t = 9.06 \, cm/s \cdot 10s = 90.6 \, cm ]For y component:
[ v{y 0} = v0 \cdot \sin(25°) = 10 \cdot \sin(25°) = 4.23 \, cm/s ]
[ \Delta y = v_y \cdot t = 4.23 \, cm/s \cdot 10s = 42.3 \, cm ]
Components of Motion
To analyze any vector ( B ) of magnitude ( 5 \, units ) directed at an angle ( \theta = 53° ):
Break it down using the relationships:
( v_x = B \cdot \cos(53°) = 5 \cdot \cos(53°) = 3 \, units )
( v_y = B \cdot \sin(53°) = 5 \cdot \sin(53°) = 4 \, units )
Magnitude of vector:
( |v| = \sqrt{vx^2 + vy^2} )
Conclusion: Kinematics in Two Dimensions
Kinematic problems can be simplified by resolving vectors into components and analyzing them using:
Three constant acceleration equations separately for the x and y directions.
Utilizing Pythagorean theorem and trigonometric functions to find the magnitude and direction of displacement, velocity, and acceleration vectors.