Fundamentals of Two-Dimensional Kinematics

Introduction

• Kinematics is the study of motion without considering the forces causing it.

• Two-dimensional kinematics deals with motion in a plane, considering both horizontal and vertical components.

3.1 Displacement, Velocity, and Acceleration

• Initial position = $r_{0}$

• Final position = $r$

• Displacement = $\Delta r=r-r_{0}$

• Average velocity is displacement divided by the elapsed time.

  • $\overrightarrow{\overline{v}}=\dfrac{\Delta r}{\Delta t}$

• Instantaneous velocity is how fast a car moves and the direction of motion at each instant of time.

  • $\overrightarrow{\overline{v}}=\lim _{\Delta t\rightarrow 0}\dfrac{\Delta r}{\Delta t}$

• Definition of average acceleration

  • $\overrightarrow{\overline{a}}=\dfrac{\Delta v}{\Delta t}$

3.2 Equations of Kinematics in Two Dimensions

• Kinematic equations are separated into x and y components:

• If velocity and acceleration go in the same directions, velocity increases

  • Different directions means velocity is decreasing

• When solving a problem for kinematics in two dimensions, list the given variables.

• Ensure you include proper signs.

• Variable table:

  	x-component	y-component	Units
  Displacement	$x$	$y$	m
  Initial Velocity	$v_{0x}$	$v_{0y}$	$m/s$
  Final Velocity	$v_{x}$	$v_{y}$	$m/s$
  Acceleration	$a_{x}$	$a_{y}$	$m/s²$
  Time (same for both)	$t$	$t$	s

Reasoning Strategy: Applying the Equations of Kinematics in Two Dimensions

1.) Make a drawing

2.) Decide which directions are positive and negative for both the x axis and the y axis.

3.) Write down the values that are given for any of the five kinematic variables associated with each direction.

4.) In an organized way, write down the values (with appropriate + and - signs) that are given for any of the five kinematic variables associated with the x direction and the y direction. Be on the alert for implied data, such as the phrase “starts from rest”, which means that the values of the initial velocity components are zero: vox/y = 0 m/s. The data summary boxes used in the examples are a good way of keeping track of this information. In addition, identify the variables that you are being asked to determine.

5.) Before attempting to solve a problem, verify that the given information contains values for at least three of the kinematic variables. Do this for the X and the Y direction of the motion. Once three known variables are identified, use the correct equation.

3.3: Projectile Motion

• Under the influence of gravity alone, an object near the surface of earth will acceleration downwards at 9.80 m/s²

• a_y = -9.80 m/s²

• a_x = 0

• v_x = v_0x = constant

Projectile Motion

Full Projectile

Projectile Motion (Extra Notes)

• Projectile motion refers to the motion of an object that is launched into the air and moves along a curved path under the influence of gravity.

• It occurs when an object is given an initial velocity and then moves freely under the force of gravity.

• The path followed by the object is called a parabola.

• The motion can be divided into two independent components: horizontal and vertical motion.

• The horizontal motion is constant and unaffected by gravity, while the vertical motion is influenced by gravity.

• The object experiences a constant horizontal velocity throughout its motion.

• The vertical motion is influenced by the acceleration due to gravity, which causes the object to accelerate downward.

• The time of flight is the total time taken by the object to complete its trajectory and is determined by the initial velocity and the angle of projection.

• The maximum height reached by the object is determined by the initial velocity and the angle of projection.

• The range of the projectile is the horizontal distance covered by the object and is determined by the initial velocity and the angle of projection.

• The range is maximum when the angle of projection is 45 degrees.

• The velocity of the object at any point in its trajectory can be determined by resolving the initial velocity into its horizontal and vertical components.

• The horizontal velocity remains constant, while the vertical velocity changes due to the acceleration due to gravity.

• The time taken to reach the maximum height is equal to the time taken to return to the same vertical position.

• Projectile motion is commonly observed in sports such as basketball, baseball, and javelin throwing.

3.4: Relative Velocity

• Relative velocity is the velocity of an object or observer with respect to another object or observer.

  • Equation for relative velocity: $\overrightarrow{v}_{AC}=\overrightarrow{v}_{AB}+\overrightarrow{v}_{BC}$

• It can be calculated by subtracting or adding the velocities of the objects involved, depending on their direction of motion.

• This concept is important in physics, engineering, and navigation, as it helps understand the motion of objects in relation to each other and predict their interactions.

• It is also used in solving problems related to collisions, projectile motion, and relative motion of objects in different frames of reference.

• However, it's important to note that relative velocity depends on the choice of reference frame and different observers may have different relative velocities between the same objects.

• Relative velocity is closely related to the concept of relative motion, which involves studying the motion of objects in relation to each other.

Conclusion

• Two-dimensional kinematics involves analyzing motion in a plane, considering both horizontal and vertical components.

• Displacement, velocity, and acceleration are vector quantities with magnitude and direction.

• Projectile motion follows a parabolic path