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

K12 Standards

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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

The motion of a projectile along the horizontal and vertical directions can be analyzed independently by using the equations of motion. Putting ay as minus g and ax as zero for a projectile, the equations simplify.
The initial velocity components obtained by using trigonometric relations further simplify the motion equations.
At the zenith, vy is zero. By substituting vy in the velocity equation for y-direction, time for half-flight is obtained. The zenith height of the projectile can be obtained by substituting the expression for the half-flight time in the displacement equation for y-direction.
The total duration of the flight is twice the time of half-flight. By substituting the total time in the displacement equation in the x-direction, the equation for the maximum range of the projectile is obtained.
The trajectory describes the projectile&#39;s path in a 2-dimensional space. Its equation can be obtained by using the position equations.
Rearranging the terms in x-component equation and substituting it in the equation of y-position, we get the final trajectory equation. This equation represents a parabola.

projectile motion: equations

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 into the pool, all undergo projectile motion.
Any projectile motion problem can be solved by using the following strategy:
<ol>
	<li>Initially, resolve the motion into horizontal and vertical components along the x- and y-axes. The magnitudes of the components of displacement along these axes are x and y. The magnitudes of the vertical and horizontal components of velocity are given by v&#8901;sin(&#952;) and v&#8901;cos(&#952;), respectively, where v is the magnitude of the velocity and &#952; is its direction relative to the horizontal.</li>
	<li>Treat the motion as two independent one-dimensional motions: one horizontal and one vertical. Use the kinematic equations for horizontal and vertical motion presented earlier.</li>
	<li>Solve for the unknowns in the two separate motions. Note that the only common variable between the motions is time, t. The problem-solving procedures here are the same as those for one-dimensional kinematics.</li>
	<li>Recombine quantities in the horizontal and vertical directions to find the total displacement and velocity. Finally, solve for the magnitude and direction of the displacement and velocity using</li>
</ol>
<img alt="Static equilibrium; diagram of forces &#931;Fx=0, &#931;Fy=0, torque MA=0; balance beam illustration." src="/files/ftp_upload/12635/12635_Equation_1.png" style="text-align: center; height: 16px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium; &#931;Fx=0, &#931;Fy=0, MA=0; diagram with force vectors; physics education." />
<img alt="Chromatography column diagram; protein purification process; lab equipment; separation method." src="/files/ftp_upload/12635/12635_Equation_2.png" style="text-align: center; height: 26px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium diagram with vector forces &#931;Fx=0, &#931;Fy=0, illustrating balance conditions." />
<img alt="Static equilibrium: &#931;Fx=0, &#931;Fy=0, MA=0; diagram with force vectors for structural analysis." src="/files/ftp_upload/12635/12635_Equation_3.png" style="text-align: center; height: 19px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium equation diagram; demonstrates &#931;Fx=0, &#931;Fy=0 conditions in force balance." />
where &#952; is the direction of the displacement.
This text is adapted from <a href="https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion">Openstax, University Physics Volume 1, Section 4.3: Projectile Motion.</a>

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 ...

Education

JoVE Core

Unit 1

Motion in Two or Three Dimensions

KEY TERMS AND CONCEPTS

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...

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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.

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PS.Phy.1.1

Position and Displacement Vectors

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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 ...

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

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Non-uniform Circular Motion

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Relative Velocity in One Dimension

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Relative Velocity in Two Dimensions

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

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Free Body Diagrams: Examples

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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

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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

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First Law: Particles in Two-dimensional Equilibrium

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Second Law: Motion under Same Force

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Tension

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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

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Static and Kinetic Frictional Force

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PS.Phy.2.3

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PS.Phy.2.4

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

Solving Projectile Motion with Components

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