Chapter 3: Two Dimensional Kinematics

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.

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

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

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

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.

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

^{2}a

_{y}= -9.80 m/s^{2}a

_{x}= 0v

_{x}= v_{0x}= constant

**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.

**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.

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

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.

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

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

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

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.

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

^{2}a

_{y}= -9.80 m/s^{2}a

_{x}= 0v

_{x}= v_{0x}= constant

**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.

**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.

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