Name: parabolic paths in projectile motion
Uploaded: 2026-06-08T09:39:11.310Z
Duration: 1 min 25 s
Channel: JoVE

Projectile motion describes the path of an object launched into the air, such as a ball kicked during a soccer penalty.Assuming that air resistance is negligible and the only force acting is gravity results in a constant downward vertical acceleration while the horizontal acceleration remains zero.The projectile&#x2019;s motion is analyzed using a vector-valued velocity, which can be decomposed into horizontal and vertical components using the launch angle alpha.In the horizontal direction, the acceleration component is zero. Integrating the horizontal acceleration component once yields a constant horizontal velocity component, and integrating again gives the horizontal displacement.In the vertical direction, the acceleration component is constant due to gravity. Integrating once and applying the initial vertical velocity component gives the vertical velocity, and integrating a second time gives the vertical displacement.Since the horizontal displacement is linear in time and the vertical displacement is quadratic in time, the trajectory of the projectile is parabolic. Eliminating time between the parametric equations confirms this result.

projectile motion

Projectile motion models the flight of an object launched into the air, such as a soccer ball kicked during a penalty, under the simplifying assumption that air resistance is negligible. When gravity is the only force, the object experiences a steady downward acceleration at all times. This single fact explains why projectile motion can be analyzed as two independent motions happening simultaneously: a horizontal motion that does not speed up or slow down, and a vertical motion that continually changes because of gravity.The initial velocity of the projectile is treated as a vector pointing in the launch direction. Using the launch angle, this initial velocity is separated into a horizontal component and a vertical component. Gravity affects only the vertical component. As a result, the horizontal acceleration is zero, so the horizontal velocity remains constant throughout the flight. The horizontal displacement, therefore, increases at a constant rate, producing a straight-line relationship with time.In contrast, the vertical acceleration is constant and downward because of gravity. The vertical velocity decreases steadily as the projectile rises, becomes momentarily zero at the peak, and then increases in magnitude in the downward direction as the projectile falls. Because the vertical velocity changes at a constant rate, the vertical displacement follows a curved, time-squared pattern rather than a straight-line pattern.The motion is naturally described with vector-valued parametric equations by expressing the position as a function of time: one time-based expression for horizontal position and another for vertical position. The horizontal position linearly depends on time, while the vertical position depends on time in a quadratic way due to the constant acceleration of gravity. When time is eliminated between these two relationships, the remaining connection between vertical position and horizontal position is a parabola. This is why the trajectory of an ideal projectile, launched and landing under uniform gravity with negligible air resistance, forms a parabolic arc.

Projectile Motion

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Education

JoVE Core

Calculus

Unit

Vector Functions and Motion

KEY TERMS AND CONCEPTS

parabolic paths in projectile motion

Projectile motion describes the path of an object launched into the air. A soccer ball kicked during a penalty is one example. In this model, air resistance is ignored, so gravity is the only force acting on the object.

That assumption lets students treat the motion as two separate parts at the same time. The horizontal motion does not speed up or slow down, while the vertical motion keeps changing because gravity pulls downward. The initial velocity is a vector, which means it has both size and direction. Using the launch angle, that vector is split into a horizontal component and a vertical component.

Gravity affects only the vertical component. The horizontal acceleration is zero, so the horizontal velocity stays constant during the flight. That means horizontal displacement increases at a steady rate and makes a straight-line pattern with time. The vertical acceleration stays constant and downward because of gravity.

As the projectile rises, its vertical velocity decreases steadily. At the top of the path, the vertical velocity is momentarily zero. After that, the object falls, and the vertical velocity increases in the downward direction. Because the vertical velocity changes at a constant rate, the vertical displacement follows a curved time-squared pattern instead of a straight line.

Projectile motion is often written with vector-valued parametric equations. These equations give position as a function of time, with one expression for horizontal position and another for vertical position. The horizontal position depends linearly on time. The vertical position depends on time in a quadratic way because of constant gravitational acceleration.

If time is removed from those equations, the relationship between horizontal and vertical position is a parabola. That is why the path of an ideal projectile forms a parabolic arc when it is launched and lands under uniform gravity with negligible air resistance.

Parabolic Paths in Projectile Motion

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curvature and its interpretation

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Projectile motion describes the path of an object launched into the air. A soccer ball kicked during a penalty is one example. In this model, air resistance is ignored, so gravity is the only force acting on the object. That assumption lets students treat the motion as two separate parts at the same time. The horizontal motion does not speed up or slow down, while the vertical motion keeps changing because gravity pulls downward. The initial velocity is a vector, which means it has both size...

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tangential-and-normal-components-of-acceleration

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vector-functions-and-motion-problem-solving

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key terms and concepts

Explore how projectile motion splits into constant horizontal motion and gravity-driven vertical motion, creating the parabolic path of a launched object.

Parabolic Paths in Projectile Motion | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Parabolic Paths in Projectile Motion

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