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

可以通过运动学方程独立分析抛射体在水平和垂直方向上的运动。将抛射体的 ay 设为负的 g，ax 设为零后，这些方程得以简化。
利用三角关系得到的初速度分量可进一步简化运动方程。
在顶点处 vy 为零。通过代入 vy 在速度方程中 y-方向，可得到半程飞行时间。将半程飞行时间的表达式代入位移方程，可求得抛射体的最高点高度。 y方向。
飞行的总时间是半程飞行时间的两倍。将总时间代入 x 方向的位移方程，可得到抛射体最大射程的方程。
轨迹描述了抛射体在二维空间中的运动路径。其方程可通过位置方程获得。
重新整理 x 分量方程中的项，并将其代入 y 坐标方程中，可得到最终的轨迹方程。该方程表示一条抛物线。

抛体运动：方程

抛体运动在我们的日常生活中很常见。 例如，球员投掷的篮球、弓箭射出的箭、孩子们跳进游泳池，都会经历抛体运动。 任何抛体运动问题都可以参照以下步骤解决： <ol> <li>首先，将运动解析为沿 x 和 y 轴的水平和垂直分量。 沿这些轴的位移分量的大小为 x 和 y。 速度的垂直和水平分量的大小由 v⋅sin(θ) 和 v⋅cos(θ) 给出，其中 v 表示速度的大小，θ 表示速度与水平方向的夹角。</li> <li>将运动视为两个独立的一维运动：一个水平运动，一个垂直运动。 使用之前介绍的水平和垂直运动的运动学方程分别处理。</li> <li>求解两个独立动议中的未知量。 需要注意的是，两个运动之间唯一的共同变量是时间，t。 这里的求解方法与一维运动学的解题过程相同。</li> <li>重新组合水平和垂直方向的量即可算出总位移和速度。 最后，使用以下公式求解位移和速度的大小及其方向</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;"> 其中θ表示位移的方向。 本文改编自 <a href="https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion">Openstax，《大学物理》第 1 卷，第 4.3 节：抛体运动。</a>

抛体运动在我们的日常生活中十分常见。例如，球员投出的篮球、弓射出的箭，以及孩子跳跃……

Education

教育

JoVE Core

物理学

Unit 1

第一单元

Motion in Two or Three Dimensions

二维或三维运动

KEY TERMS AND CONCEPTS

关键术语与概念

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

The position of an object defines its location relative to a convenient frame of reference at any particular time. A frame of reference is an arbitrary ...

Average Velocity

To calculate the other physical quantities in kinematics, we must introduce the time variable. The time variable allows us not only to state the position ...

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

Instantaneous velocity is the quantity that measures how fast an object is moving along its path. In other words, the instantaneous velocity of an object ...

Average Acceleration

The importance of understanding acceleration spans our day-to-day experiences, as well as the vast reaches of outer space and the tiny world of subatomic ...

Instantaneous Acceleration

Acceleration is in the direction of the change in velocity, but it is not always in the direction of motion. When an object slows down, its acceleration ...

Kinematic Equations - I

When an object moves with constant acceleration, the velocity of the object changes at a constant rate throughout the motion. The kinematic equations of ...

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

The first two kinematic equations have time as a variable, but the third kinematic equation is independent of time. This equation expresses final velocity ...

Kinematic Equations: Problem Solving

When analyzing one-dimensional motion with constant acceleration, the problem-solving strategy involves identifying the known quantities and choosing the ...

Free-falling Bodies: Introduction

All objects, neglecting air resistance, fall with the same acceleration towards the Earth&#39;s center due to the force exerted by the Earth&#39;s ...

Free-falling Bodies: Example

An object falling without any air resistance under the influence of gravitational force is said to be in free-fall. For free-falling bodies, the ...

velocity-and-position-by-graphical-method

Velocity and Position by Graphical Method

Velocity and position can be calculated from the known function of acceleration as a function of time. The total area under the acceleration-time graph ...

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

To calculate other physical quantities in kinematics, the time variable must be introduced. The time variable not only allows us to state where an object ...

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 ...

Direction of Acceleration Vectors

Acceleration occurs when velocity changes in magnitude (an increase or decrease in speed), direction, or both. Although acceleration is in the direction ...

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

Principle of Linear Impulse and Momentum for a Single Particle: Problem Solving

Consider a massless pulley system consisting of a wooden box and a cylinder of known masses. Determine the speed of the attached objects at a specific ...

Non-uniform Circular Motion

In&#160;uniform circular motion, the particle executing circular motion has a constant speed, and the circle is at a fixed radius. However, not all ...

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

In most situations, forces can be grouped into two categories: contact forces and field forces.&#160; Contact forces occur as a result of direct physical ...

Drawing Free-body Diagrams: Rules

The first step in describing and analyzing most phenomena in physics involves the careful drawing of a free-body diagram. Free-body diagrams are useful in ...

Free Body Diagrams: Examples

Solving problems that involve forces is easy using free-body diagrams. A free-body diagram is a sketch showing all the external forces that are acting on ...

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

Experience suggests that an object at rest remains at rest if left alone, and that an object in motion tends to slow down and stop unless some effort is ...

Internal and External Forces

Newton&#39;s first law states that a net external force causes a change in motion. External forces act on an object or system, originating outside of the ...

Newton's Second Law

Newton&#39;s second law is closely related to his first law of motion. It mathematically gives the cause-and-effect relationship between force and changes ...

Mass and Weight

Mass and weight are often used interchangeably in everyday conversation. For example,&#160; medical records often show our weight in kilograms, but never ...

Weightlessness

When an object is dropped, it accelerates toward the center of the Earth. If the net external force on the object is its weight, it is said to be in free ...

Apparent Weight

True weight is the measure of the gravitational force acting on an object. However, if the object accelerates, its measured weight is different from its ...

/files/media/science-education/science-education-thumbs/14695.jpg

Newton's Third Law: Introduction

Whenever one body exerts a force on a second body, the first body experiences a force equal in magnitude and opposite in direction, to the force that it ...

Newton's Third Law: Examples

Newton&#39;s third law states that every action has an equal and opposite reaction. Consider a swimmer pushing off the side of a pool. They push against ...

first-law-particles-in-one-dimensional-equilibrium

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 ...

Second Law: Motion under Same Force

Newton&#39;s laws can be applied to bodies at rest and bodies in motion. Newton&#39;s first law is applied to bodies in equilibrium, whereas the second ...

second-law-motion-under-same-acceleration

Second Law: Motion under Same Acceleration

Newton&#39;s second law of motion applies to bodies moving under the same acceleration. For example, when a baggage tractor pulls luggage carts, each cart ...

Drag Force and Terminal Speed

An interesting force in everyday life is the force of drag on an object when it is moving in a fluid. Like friction, the drag force always opposes the ...

/files/media/science-education/science-education-thumbs/14697.jpg

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; ...

/files/media/science-education/science-education-thumbs/14698.jpg

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 ...

Dynamics of Circular Motion

An object undergoing circular motion, like a race car, is accelerating because it is changing the direction of its velocity. This centrally directed ...

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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