Name: solving projectile motion with components
Uploaded: 2025-05-22T09:05:40.000Z
Duration: 1 min 26 s
Channel: JoVE

K12 Standards

north-carolina-k-12-science-standard-course-of-study

North Carolina K-12 Science Standard Course of Study

9-12

Physics

motion-and-stability-forces-and-interactions

Motion and Stability- Forces and Interactions

PS.Phy.1.2

Projectile Motion: Equations

Die Bewegung eines Projektils entlang der horizontalen und vertikalen Richtung kann unabhängig voneinander mit Hilfe der Bewegungsgleichungen analysiert werden. Wenn man einy als minus g und einx als Null für ein Projektil setzt, vereinfachen sich die Gleichungen.
Die anfänglichen Geschwindigkeitskomponenten, die durch die Verwendung trigonometrischer Beziehungen erhalten wurden, vereinfachen die Bewegungsgleichungen weiter.
Im Zenit ist vy Null. Durch Einsetzen der y-Richtung durch vy in der Geschwindigkeitsgleichung erhält man die Zeit für den Halbflug. Die Zenithöhe des Projektils kann ermittelt werden, indem die y-Richtung durch den Ausdruck für die halbe Flugzeit in der Verschiebungsgleichung ersetzt wird.
Die Gesamtdauer des Fluges ist doppelt so lang wie die Zeit des halben Fluges. Durch Einsetzen der Gesamtzeit in der Verschiebungsgleichung in x-Richtung erhält man die Gleichung für die maximale Reichweite des Projektils.
Die Flugbahn beschreibt den Weg des Projektils in einem 2-dimensionalen Raum. Seine Gleichung kann mit Hilfe der Positionsgleichungen erhalten werden.
Wenn wir die Terme in der x-Komponentengleichung neu anordnen und in der Gleichung der y-Position einsetzen, erhalten wir die endgültige Trajektoriengleichung. Diese Gleichung stellt eine Parabel dar.

wurfbewegung: gleichungen

Die Wurfbewegung wird in unserem täglichen Leben häufig beobachtet. Zum Beispiel unterliegen ein von einem Spieler geworfener Basketball, ein aus einem Bogen abgeschossener Pfeil und Kinder, die in den Pool springen, alle einer Wurfparabel.

Jedes Problem der Wurfbewegung kann mit der folgenden Strategie gelöst werden:

<ol style="list-style-type: disc;">
	<li>Zerlegen Sie zuerst die Bewegung in horizontale und vertikale Komponenten entlang der x- und y-Achsen. Die Beträge der Verschiebungskomponenten entlang dieser Achsen werden mit x und y bezeichnet. Die Beträge der vertikalen und horizontalen Geschwindigkeitskomponenten werden durch v⋅sin(θ) bzw. v⋅cos(θ) beschrieben, wobei v der Geschwindigkeitsbetrag ist und θ seine Richtung relativ zur Horizontalen ist.</li>
	<li>Behandeln Sie die Bewegung als zwei unabhängige eindimensionale Bewegungen: eine horizontale und eine vertikale. Verwenden Sie die kinematischen Gleichungen für horizontale und vertikale Bewegungen, die zuvor präsentiert wurden.</li>
	<li>Lösen Sie die Unbekannten in den beiden separaten Bewegungen. Beachten Sie, dass die einzige gemeinsame Variable zwischen den Bewegungen die Zeit t ist. Die problemlösenden Verfahren sind hier dieselben wie bei der eindimensionalen Kinematik.</li>
	<li>Kombinieren Sie die Größen in horizontaler und vertikaler Richtung, um die Gesamtverschiebung und -geschwindigkeit zu finden. Lösen Sie abschließend den Betrag und die Richtung der Verschiebung und Geschwindigkeit mit den folgenden Gleichungen:</li>
</ol>

<img alt="Equation1" src="/files/ftp_upload/12635/12635_Equation_1.png" style="text-align: center; height: 16px;" />

<img alt="Equation2" src="/files/ftp_upload/12635/12635_Equation_2.png" style="text-align: center; height: 26px;" />

<img alt="Equation3" src="/files/ftp_upload/12635/12635_Equation_3.png" style="text-align: center; height: 19px;" />

wobei θ die Richtung der Verschiebung ist.

Dieser Text ist angepasst von <a href="https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion">Openstax, University Physics Volume 1, Abschnitt 4.3: Wurfbewegung.</a>

Wurfbewegung: Gleichungen

Projektilbewegungen sind in unserem täglichen Leben häufig zu beobachten. Zum Beispiel ein Basketball, der von einem Spieler geworfen wird, ein Pfeil, der von einem Bogen abgeschossen wird, und Kinder, die springen ...

Education

Lehre

JoVE Core

Physik

Unit 1

Einheit 1

Motion in Two or Three Dimensions

Bewegung in zwei oder drei Dimensionen

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

solving projectile motion with components

Projectile motion can be solved by breaking the motion into horizontal and vertical components. This approach is used for common examples like a basketball throw, an arrow shot from a bow, and kids jumping into a pool. The motion is then analyzed along the x-axis and y-axis.

Start by separating displacement into x and y parts. The horizontal and vertical parts of velocity are also separated. If the projectile has speed v and direction θ relative to the horizontal, the vertical component is v·sin(θ) and the horizontal component is v·cos(θ).

Next, treat the horizontal motion and the vertical motion as two independent one-dimensional motions. Use the kinematic equations for each direction, just as in earlier one-dimensional kinematics problems. The key shared variable is time, t, so solve each part separately and use time to connect them.

After finding the unknowns, recombine the horizontal and vertical quantities to get the total displacement and velocity. Then use the combined x and y values to find the magnitude and direction of displacement and velocity. In this step, θ is the direction of the displacement.

Solving Projectile Motion with Components

solving-projectile-motion-with-components

position and displacement vectors

Describing Motion with Position Changes

Describing motion starts with position. Position tells where an object is at a particular time. It must be measured relative to a frame of reference, which is an arbitrary set of axes used to describe position and motion. Earth is often used as a frame of reference. It helps describe an object’s location in relation to stationary objects on Earth. Many important motions also take place in two dimensions, or in a plane. In those cases, position, velocity, and acceleration can each be described...

average and instantaneous velocity vectors

average-and-instantaneous-velocity-vectors

Position Change and Instant Velocity

Position change and instant velocity are key ideas in kinematics. To describe motion, time must be included. Time tells us where an object is and how fast it is moving. The speed of an object comes from how its position changes with time. Each position is tied to a specific time. When the details of motion at each instant are not important, the rate is given as average velocity. Average velocity is a vector. It equals the total displacement between two points divided by the elapsed time. Real...

acceleration vectors

How Velocity Changes Create Acceleration

Acceleration describes how velocity changes over time. In physics, acceleration is a vector, which means it has both size and direction. It points in the same direction as the change in velocity, written as Δv. The larger the acceleration, the larger the change in velocity over a given time. Because velocity is a vector, it can change in more than one way. An object can speed up, slow down, change direction, or do both at once. A runner traveling 10 km/h due east who stops, turns around, and...

direction of acceleration vectors

Negative Acceleration and Motion Reversal

Acceleration describes a change in velocity. That change can be a change in speed, a change in direction, or both. The acceleration vector points in the direction of the velocity change, not always in the direction the object is moving. When an object slows down, its acceleration points opposite to its motion. This is often called deceleration. The term can be confusing in physics because deceleration is not a vector. It does not give a direction relative to a coordinate system, so the correct...

projectile motion

Projectile Paths Under Gravity

Projectile motion describes the path of an object thrown into the air under Earth’s gravity. The object is called a projectile, and its curved path is called the trajectory. This motion appears in many real situations, such as fireworks, a golf ball in flight, meteors entering Earth’s atmosphere, and bullets being fired. When a falling object has no horizontal movement, it is described as one-dimensional projectile motion. When motion includes both horizontal and vertical parts, it is...

/files/media/science-education/science-education-thumbs/12634.jpg

projectile motion: equations

Projectile motion can be solved by breaking the motion into horizontal and vertical components. This approach is used for common examples like a basketball throw, an arrow shot from a bow, and kids jumping into a pool. The motion is then analyzed along the x-axis and y-axis. Start by separating displacement into x and y parts. The horizontal and vertical parts of velocity are also separated. If the projectile has speed v and direction θ relative to the horizontal, the vertical component is...

projectile motion: example

Projectile Motion in Sports and Orbit

Projectile motion helps explain both sports throws and motion near Earth. A javelin thrower uses this idea to send the javelin as far as possible. The thrower also takes a short run-up to increase the initial speed of the javelin. For a projectile on level ground, the range is the horizontal distance it travels. The range is largest at a 45° angle, so javelin throwers try to release the javelin as close to that angle as possible. In this case, the range equation assumes the distance is very...

uniform circular motion

Centripetal Acceleration in Circular Motion

Uniform circular motion describes motion in a circle at constant speed. An object follows a curved path, but its speed does not change. The motion of a point on a propeller spinning at a steady rate and the second, minute, and hour hands of a watch are common examples. Even with constant speed, these objects are still accelerating. That idea can seem surprising because in one-dimensional motion, constant speed means zero acceleration. In two- and three-dimensional motion, a particle can...

non-uniform circular motion

Circular Motion with Changing Speed

Circular motion can happen with changing speed. In this case, the object still follows a circular path, but it may speed up or slow down as it moves. When speed changes during circular motion, the motion is called non-uniform circular motion. This adds tangential acceleration, which points along the path of motion and shows a change in speed. Centripetal acceleration is still present in circular motion. It describes the time rate of change of the direction of the velocity vector and points...

relative velocity in one dimension

Relative Motion in a Train and Earth

Relative motion depends on the reference frame used to measure velocity. A reference frame is the point of view from which motion is described. In many examples, Earth is the reference frame. If someone is said to be sitting in a train moving east at 10 m/s, that speed is measured relative to Earth.

relative velocity in two dimensions

Adding Velocities Across Reference Frames

Relative velocity describes motion as measured from a chosen reference frame. It can also describe the motion of one reference frame compared with another. In two dimensions, this idea helps track how position and velocity change from frame to frame. Consider a particle P and two reference frames, S and S′. The position of the origin of S′, measured in S, and the position of P, measured in S′, can be combined with vector addition to find the position of P in S. This relation shows how motion...

key terms and concepts

Learn how to solve projectile motion by splitting motion into horizontal and vertical components, using kinematic equations, and recombining displacement and velocity.

Solving Projectile Motion with Components | JoVE K12

physics

JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

ohios-learning-standards-and-model-curriculum

Ohio’s Learning Standards and Model Curriculum

Motion

Projectile Motion

Tennessee Academic Standards

High School

Motion and Stability: Forces and Interactions

PHYS.PS2.4

PHYS.PS2.13

Virginia  Standards of Learning

HIgh School

Physical Science

PS.8

PS.8.a

PH.2

PH.2.d

Texas Essential Knowledge and Skills

PHYS.5

PHYS.5.D

Arizona Science Standards

HS+Phy.P3

HS+Phy.P3U1.3

Alabama Course of Study Science

9-12.P.1.1.c

Florida Department of Education

SC.912.P.12

SC.912.P.12.2

Georgia Standards of Excellence

SP1.

Position and Displacement

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Average Velocity

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Instantaneous Velocity - I

The average velocity during a time interval cannot tell us how fast or in what direction a particle is moving at any given time during the interval. To ...

Instantaneous Velocity - II

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Average Acceleration

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Instantaneous Acceleration

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Kinematic Equations - I

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Kinematic Equations - II

The second kinematic equation expresses the final position of an object in terms of its initial position, the distance traveled with the initial constant ...

Kinematic Equations - III

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Velocity and Position by Graphical Method

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Velocity and Position by Integral Method

If acceleration as a function of time is known, then velocity and position functions can be derived using integral calculus. For constant acceleration, ...

Use models (graphs, equations, diagrams) to infer motion in one dimension.

PS.Phy.1.1

Position and Displacement Vectors

To describe the motion of an object, one should first be able to describe its position (where it is at any particular time). More precisely, the position ...

Average and Instantaneous Velocity Vectors

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Acceleration Vectors

In everyday conversation, accelerating means speeding up. Acceleration is a vector in the same direction as the change in velocity, &#916;v, therefore the ...

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An object thrown in the air follows a parabolic path under the influence of Earth&#39;s gravitational force. The motion of such an object is called ...

Projectile motion is commonly observed in our day-to-day life. For example, a basketball thrown by a player, an arrow shot from a bow, and kids jumping ...

Projectile Motion: Example

The theory of projectile motion is very useful for players of several sports to improve their performance. For example, a javelin thrower needs to throw ...

principle-linear-impulse-momentum-for-single-particle-problem

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Non-uniform Circular Motion

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Relative Velocity in One Dimension

The understanding of the concept of reference frames is essential to discuss relative motion in one or more dimensions. When we say that an object has a ...

Relative Velocity in Two Dimensions

Relative velocity is the velocity of an object as observed from a particular reference frame, or the velocity of one reference frame with respect to ...

Use models (graphs, equations, diagrams) to infer motion in two dimensions.

Force

Forces affect every moment of our life. Our bodies are held to the Earth by force, and they are held together by the forces of charged particles. When we ...

Types of Forces

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Drawing Free-body Diagrams: Rules

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Use free body models to qualitatively and quantitatively analyze systems of forces in one dimension and two dimensions.

PS.Phy.2.1

Newton's First Law: Introduction

Motion draws our attention. Motion itself can be beautiful, causing us to marvel at the forces needed to create spectacular sights, such as that of a ...

Newton's First Law: Application

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Internal and External Forces

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Newton's Second Law

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First Law: Particles in One-dimensional Equilibrium

Newton&#39;s first law of motion states that a body at rest remains at rest, or if in motion, remains in motion at constant velocity, unless acted on by a ...

first-law-particles-in-two-dimensional-equilibrium

First Law: Particles in Two-dimensional Equilibrium

Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic ...

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Second Law: Motion under Same Acceleration

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Drag Force and Terminal Speed

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Tension

Tension is a force along the length of a medium, in particular, a force carried by a flexible medium, such as a rope or cable. The word &#34;tension&#34; ...

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Carry out investigations to explain the interactions of forces on an object according to Newton's Laws of Motion.

PS.Phy.2.2

Frictional Force

When a body is in motion, it encounters resistance because the body interacts with its surroundings. This resistance is known as friction, a common yet ...

Static and Kinetic Frictional Force

One of the simpler characteristics of sliding friction is that it is parallel to the contact surfaces between systems, and is always in a direction that ...

Use models to qualitatively and quantitatively analyze basic forces related to movement of an object in a circular path (centripetal force).

PS.Phy.2.3

Analyze the motion of objects using time, distance, displacement, speed, velocity, and acceleration.

Solving Projectile Motion with Components

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