Storyboard

Objects can intersect when they coincide in angle at the same moment. To achieve this, they must move from their respective initial angles with angular velocities that allow them to coincide in angle and time at the end of the journey.

>Model

ID:(1450, 0)

Iframe

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ID:(15411, 0)

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In the case of an intersection, two bodies are moving in such a way that they will meet at angle of intersection ($\theta$) at time a intersection time ($t$).

To achieve this, each body:

• Begins its displacement at the initial time of first object ($t_1$) at the initial angle of the first body ($\theta_1$) with a angular velocity of body 1 ($\omega_1$).
• Begins its displacement at the initial time of second object ($t_2$) at the initial angle of the second body ($\theta_2$) with a angular velocity of body 2 ($\omega_2$).

These conditions must be met to achieve the intersection.

Thus, the angle over time diagrams can be overlaid as shown in the following representation:

ID:(15517, 0)

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In the case of an intersection or collision between two objects, it's common for the angular velocity of body 1 ($\omega_1$) and the angular velocity of body 2 ($\omega_2$) to be configured such that they coincide.

This means that the angle traveled by the first body ($\Delta\theta_1$) and the travel time of first object ($\Delta t_1$) must result in a angular velocity of body 1 ($\omega_1$),

so that, with the angle traveled by the second body ($\Delta\theta_2$) and the travel time of second object ($\Delta t_2$), we get a angular velocity of body 2 ($\omega_2$),

so that they finally coincide in time and space (position):

ID:(15516, 0)

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In the case of a movement where two objects intersect, such as the angle of intersection ($\theta$) and the intersection time ($t$), it is common for both. Therefore, if for the first object, the initial time of first object ($t_1$) and the initial angle of the first body ($\theta_1$) with the angular velocity of body 1 ($\omega_1$) fulfill:

and for the second object, the initial time of second object ($t_2$) and the initial angle of the second body ($\theta_2$) with the angular velocity of body 2 ($\omega_2$) fulfill:

which is represented as:

ID:(15518, 0)

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Parameters

$r$

r

Radius

m

Variables

$\theta$

theta

Angle of intersection

rad

$\Delta\theta_1$

Dtheta_1

Angle traveled by the first body

rad

$\Delta\theta_2$

Dtheta_2

Angle traveled by the second body

rad

$\omega_1$

omega_1

Angular velocity of body 1

rad/s

$\omega_2$

omega_2

Angular velocity of body 2

rad/s

$\Delta\theta$

Dtheta

Difference of Angles

rad

$\theta_1$

theta_1

Initial angle of the first body

rad

$\theta_2$

theta_2

Initial angle of the second body

rad

$t_1$

t_1

Initial time of first object

s

$t_2$

t_2

Initial time of second object

s

$t$

t

Intersection time

s

$v_1$

v_1

Speed of the first object

m/s

$v_2$

v_2

Speed of the second object

m/s

$\Delta t_1$

Dt_1

Travel time of first object

s

$\Delta t_2$

Dt_2

Travel time of second object

s

Calculations

• Alternative method
• Traditional method
• Strategy
• 2D
• 3D

Equation

Number

First, select the equation: to , then, select the variable: to

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Calculations

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable Given Calculate Target : Equation To be used

Equations

ID:(15422, 0)

Equation

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To describe the rotation of an object, we need to determine the angle variation ($\Delta\theta$). This is achieved by subtracting the initial Angle ($\theta_0$) from the angle ($\theta$), which is reached by the object during its rotation:

$\Delta\theta = \theta - \theta_1$

$\Delta\theta = \theta - \theta_0$

Conditions

Variables

$\theta$

$\theta$

Angle of intersection

$rad$

10307

$\Delta\theta$

Difference of Angles

$rad$

5299

$\theta_0$

$\theta_1$

Initial angle of the first body

$rad$

10308

Relation

ID:(3680, 1)

Equation

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To describe the rotation of an object, we need to determine the angle variation ($\Delta\theta$). This is achieved by subtracting the initial Angle ($\theta_0$) from the angle ($\theta$), which is reached by the object during its rotation:

$\Delta\theta = \theta - \theta_2$

$\Delta\theta = \theta - \theta_0$

Conditions

Variables

$\theta$

$\theta$

Angle of intersection

$rad$

10307

$\Delta\theta$

Difference of Angles

$rad$

5299

$\theta_0$

$\theta_2$

Initial angle of the second body

$rad$

10309

Relation

ID:(3680, 2)

Equation

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To describe the motion of an object, we need to calculate the time elapsed ($\Delta t$). This magnitude is obtained by measuring the start Time ($t_0$) and the the time ($t$) of said motion. The duration is determined by subtracting the initial time from the final time:

$\Delta t_1 \equiv t - t_1$

$\Delta t \equiv t - t_0$

Conditions

Variables

$t_0$

$t_1$

Initial time of first object

$s$

10252

$t$

$t$

Intersection time

$s$

10259

$\Delta t$

$\Delta t_1$

Travel time of first object

$s$

10256

Relation

ID:(4353, 1)

Equation

>Top, >Model

To describe the motion of an object, we need to calculate the time elapsed ($\Delta t$). This magnitude is obtained by measuring the start Time ($t_0$) and the the time ($t$) of said motion. The duration is determined by subtracting the initial time from the final time:

$\Delta t_2 \equiv t - t_2$

$\Delta t \equiv t - t_0$

Conditions

Variables

$t_0$

$t_2$

Initial time of second object

$s$

10253

$t$

$t$

Intersection time

$s$

10259

$\Delta t$

$\Delta t_2$

Travel time of second object

$s$

10257

Relation

ID:(4353, 2)

Equation

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In the case where the angular velocity is constant, the mean angular velocity ($\bar{\omega}$) coincides with the value of the initial Angular Speed ($\omega_0$), so

In this scenario, we can calculate the angle traveled as a function of time by recalling that it is associated with the difference between the current and initial angles, as well as the current and initial time. Therefore, the angle ($\theta$) is equal to the initial Angle ($\theta_0$), the initial Angular Speed ($\omega_0$), the time ($t$), and the start Time ($t_0$) as shown below:

$\theta = \theta_1 + \omega_1 ( t - t_1 )$

$\theta = \theta_0 + \omega_0 ( t - t_0 )$

Conditions

Variables

$\theta$

$\theta$

Angle of intersection

$rad$

10307

$\theta_0$

$\theta_1$

Initial angle of the first body

$rad$

10308

$\omega_0$

$\omega_1$

Angular velocity of body 1

$rad/s$

10312

$t_0$

$t_1$

Initial time of first object

$s$

10252

$t$

$t$

Intersection time

$s$

10259

Relation

Development

In the case where the initial Angular Speed ($\omega_0$) is equal to the mean angular velocity ($\bar{\omega}$),

$\bar{\omega} = \omega_0$

Therefore, with the difference of Angles ($\Delta\theta$), which is equal to the angle ($\theta$) divided by the initial Angle ($\theta_0$), we obtain:

$\Delta\theta = \theta - \theta_0$

And with the time elapsed ($\Delta t$), which is equal to the time ($t$) divided by the start Time ($t_0$), we obtain:

$\Delta t \equiv t - t_0$

We can rewrite the equation for the mean angular velocity ($\bar{\omega}$) as:

$\bar{\omega} \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$

This can be expressed as:

$\omega_0 = \omega = \displaystyle\frac{\Delta\theta}{\Delta t} = \displaystyle\frac{\theta - \theta_0}{t - t_0}$

Solving for it, we get:

$\theta = \theta_0 + \omega_0 ( t - t_0 )$

The equation represents a straight line in angle-time space.

ID:(1023, 1)

Equation

>Top, >Model

In the case where the angular velocity is constant, the mean angular velocity ($\bar{\omega}$) coincides with the value of the initial Angular Speed ($\omega_0$), so

In this scenario, we can calculate the angle traveled as a function of time by recalling that it is associated with the difference between the current and initial angles, as well as the current and initial time. Therefore, the angle ($\theta$) is equal to the initial Angle ($\theta_0$), the initial Angular Speed ($\omega_0$), the time ($t$), and the start Time ($t_0$) as shown below:

$\theta = \theta_2 + \omega_2 ( t - t_2 )$

$\theta = \theta_0 + \omega_0 ( t - t_0 )$

Conditions

Variables

$\theta$

$\theta$

Angle of intersection

$rad$

10307

$\theta_0$

$\theta_2$

Initial angle of the second body

$rad$

10309

$\omega_0$

$\omega_2$

Angular velocity of body 2

$rad/s$

10313

$t_0$

$t_2$

Initial time of second object

$s$

10253

$t$

$t$

Intersection time

$s$

10259

Relation

Development

In the case where the initial Angular Speed ($\omega_0$) is equal to the mean angular velocity ($\bar{\omega}$),

$\bar{\omega} = \omega_0$

Therefore, with the difference of Angles ($\Delta\theta$), which is equal to the angle ($\theta$) divided by the initial Angle ($\theta_0$), we obtain:

$\Delta\theta = \theta - \theta_0$

And with the time elapsed ($\Delta t$), which is equal to the time ($t$) divided by the start Time ($t_0$), we obtain:

$\Delta t \equiv t - t_0$

We can rewrite the equation for the mean angular velocity ($\bar{\omega}$) as:

$\bar{\omega} \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$

This can be expressed as:

$\omega_0 = \omega = \displaystyle\frac{\Delta\theta}{\Delta t} = \displaystyle\frac{\theta - \theta_0}{t - t_0}$

Solving for it, we get:

$\theta = \theta_0 + \omega_0 ( t - t_0 )$

The equation represents a straight line in angle-time space.

ID:(1023, 2)

Equation

>Top, >Model

To estimate the displacement of an object, it's necessary to know its the angular Speed ($\omega$) as a function of the time ($t$). Therefore, the the mean angular velocity ($\bar{\omega}$) is introduced, defined as the ratio between the angle variation ($\Delta\theta$) and the time elapsed ($\Delta t$).

To measure this, a system like the one shown in the image can be used:

To determine the average angular velocity, a reflective element is placed on the axis or on a disk with several reflective elements, and the passage is recorded to estimate the length of the arc $\Delta s$ and the angle associated with the radius $r$. Then the time difference when the mark passes in front of the sensor is recorded as $\Delta t$. The average angular velocity is determined by dividing the angle traveled by the time elapsed.

The equation that describes the average angular velocity is:

$\omega_1 \equiv\displaystyle\frac{ \Delta\theta_1 }{ \Delta t_1 }$

$\bar{\omega} \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$

Conditions

Variables

$\Delta\theta$

$\Delta\theta_1$

Angle traveled by the first body

$rad$

10310

$\bar{\omega}$

$\omega_1$

Angular velocity of body 1

$rad/s$

10312

$\Delta t$

$\Delta t_1$

Travel time of first object

$s$

10256

Relation

Development

The definition of the mean angular velocity ($\bar{\omega}$) is considered as the angle variation ($\Delta\theta$),

$\Delta\theta = \theta - \theta_0$

and the time elapsed ($\Delta t$),

$\Delta t \equiv t - t_0$

The relationship between both is defined as the mean angular velocity ($\bar{\omega}$):

$\bar{\omega} \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$

It should be noted that the average velocity is an estimation of the actual angular velocity. The main problem is that:

If the angular velocity varies during the elapsed time, the value of the average angular velocity can be very different from the average angular velocity.

Therefore, the key is:

Determine the velocity in a sufficiently short elapsed time to minimize its variation.

ID:(3679, 1)

Equation

>Top, >Model

To estimate the displacement of an object, it's necessary to know its the angular Speed ($\omega$) as a function of the time ($t$). Therefore, the the mean angular velocity ($\bar{\omega}$) is introduced, defined as the ratio between the angle variation ($\Delta\theta$) and the time elapsed ($\Delta t$).

To measure this, a system like the one shown in the image can be used:

To determine the average angular velocity, a reflective element is placed on the axis or on a disk with several reflective elements, and the passage is recorded to estimate the length of the arc $\Delta s$ and the angle associated with the radius $r$. Then the time difference when the mark passes in front of the sensor is recorded as $\Delta t$. The average angular velocity is determined by dividing the angle traveled by the time elapsed.

The equation that describes the average angular velocity is:

$\omega_2 \equiv\displaystyle\frac{ \Delta\theta_2 }{ \Delta t_2 }$

$\bar{\omega} \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$

Conditions

Variables

$\Delta\theta$

$\Delta\theta_2$

Angle traveled by the second body

$rad$

10311

$\bar{\omega}$

$\omega_2$

Angular velocity of body 2

$rad/s$

10313

$\Delta t$

$\Delta t_2$

Travel time of second object

$s$

10257

Relation

Development

The definition of the mean angular velocity ($\bar{\omega}$) is considered as the angle variation ($\Delta\theta$),

$\Delta\theta = \theta - \theta_0$

and the time elapsed ($\Delta t$),

$\Delta t \equiv t - t_0$

The relationship between both is defined as the mean angular velocity ($\bar{\omega}$):

$\bar{\omega} \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$

It should be noted that the average velocity is an estimation of the actual angular velocity. The main problem is that:

If the angular velocity varies during the elapsed time, the value of the average angular velocity can be very different from the average angular velocity.

Therefore, the key is:

Determine the velocity in a sufficiently short elapsed time to minimize its variation.

ID:(3679, 2)

Equation

>Top, >Model

If we divide the relationship between the distance traveled in a time ($\Delta s$) and the radius ($r$) by the angle variation ($\Delta\theta$),

and then divide it by the time elapsed ($\Delta t$), we obtain the relationship that allows us to calculate the speed ($v$) along the orbit, known as the tangential velocity, which is associated with the angular Speed ($\omega$):

$v_1 = r \omega_1$

$v = r \omega$

Conditions

Variables

$\omega$

$\omega_1$

Angular velocity of body 1

$rad/s$

10312

$r$

Radius

$m$

9894

$v$

$v_1$

Speed of the first object

$m/s$

10248

Relation

Development

As the mean Speed ($\bar{v}$) is with the distance traveled in a time ($\Delta s$) and the time elapsed ($\Delta t$), equal to

$\bar{v} \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$

and with the distance traveled in a time ($\Delta s$) expressed as an arc of a circle, and the radius ($r$) and the angle variation ($\Delta\theta$) are

$\Delta s=r \Delta\theta$

and the definition of the mean angular velocity ($\bar{\omega}$) is

$\bar{\omega} \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$

then,

$v=\displaystyle\frac{\Delta s}{\Delta t}=r\displaystyle\frac{\Delta\theta}{\Delta t}=r\omega$

Since the relationship is general, it can be applied for instantaneous values, resulting in

$v = r \omega$

.

ID:(3233, 1)

Equation

>Top, >Model

If we divide the relationship between the distance traveled in a time ($\Delta s$) and the radius ($r$) by the angle variation ($\Delta\theta$),

and then divide it by the time elapsed ($\Delta t$), we obtain the relationship that allows us to calculate the speed ($v$) along the orbit, known as the tangential velocity, which is associated with the angular Speed ($\omega$):

$v_2 = r \omega_2$

$v = r \omega$

Conditions

Variables

$\omega$

$\omega_2$

Angular velocity of body 2

$rad/s$

10313

$r$

Radius

$m$

9894

$v$

$v_2$

Speed of the second object

$m/s$

10249

Relation

Development

As the mean Speed ($\bar{v}$) is with the distance traveled in a time ($\Delta s$) and the time elapsed ($\Delta t$), equal to

$\bar{v} \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$

and with the distance traveled in a time ($\Delta s$) expressed as an arc of a circle, and the radius ($r$) and the angle variation ($\Delta\theta$) are

$\Delta s=r \Delta\theta$

and the definition of the mean angular velocity ($\bar{\omega}$) is

$\bar{\omega} \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$

then,

$v=\displaystyle\frac{\Delta s}{\Delta t}=r\displaystyle\frac{\Delta\theta}{\Delta t}=r\omega$

Since the relationship is general, it can be applied for instantaneous values, resulting in

$v = r \omega$

.

ID:(3233, 2)