Name: using the second kinematic equation
Uploaded: 2025-05-22T09:05:40.000Z
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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.1.1

Kinematic Equations - II

Die erste kinematische Gleichung drückt die Geschwindigkeit als Funktion der Anfangsgeschwindigkeit und der Geschwindigkeitsänderung aus.
Zur Ableitung der zweiten kinematischen Gleichung werden zwei Gleichungen für die durchschnittliche x-Geschwindigkeit während eines Intervalls von t gleich Null bis zu einem späteren Zeitpunkt t verwendet. Die erste Gleichung für die durchschnittliche x-Geschwindigkeit ist die Änderung der Verschiebung über die Zeitänderung, wie in den vorherigen Lektionen beschrieben. Hier wird die Anfangsposition mit x0 zum Zeitpunkt t gleich Null und x zum späteren Zeitpunkt t bezeichnet.
Unter der Annahme, dass die Beschleunigung konstant ist, kann die durchschnittliche x-Geschwindigkeit auch als Durchschnitt der Geschwindigkeiten zum Anfangs- und Endzeitpunkt dargestellt werden. Setzt man hier die erste kinematische Gleichung ein, erhält man eine Gleichung für die Position x als Funktion der Zeit.
Diese Gleichung zeigt, dass die Position eines Objekts zu einem beliebigen Zeitpunkt t die Summe aus seiner Anfangsposition, der bei konstanter Anfangsgeschwindigkeit zurückgelegten Strecke und der während der Geschwindigkeitsänderung zurückgelegten Strecke ist.

kinematische gleichungen - ii

Die zweite kinematische Gleichung beschreibt die Endposition eines Objekts in Bezug auf seine Anfangsposition, die mit konstanter Anfangsgeschwindigkeit zurückgelegte Strecke und die aufgrund einer Änderung der Geschwindigkeit zurückgelegte Strecke. Ähnlich wie bei der ersten kinematischen Gleichung ist auch diese Gleichung nur gültig, wenn die Beschleunigung während der Bewegung eines Objekts konstant ist.

Angenommen, ein Auto fährt auf einer 200 m langen Auffahrt in den Autobahnverkehr ein. Wenn seine Anfangsgeschwindigkeit 10 m/s beträgt und es mit 2 m/s^2 beschleunigt, kann die Zeit, die das Auto benötigt, um die 200 m lange Auffahrt zu durchfahren, mithilfe der zweiten kinematischen Gleichung berechnet werden. Dabei sind die bekannten Größen die Strecke von 200 m, die Anfangsgeschwindigkeit des Autos von 10 m/s und die Beschleunigung von 2 m/s^2. Unter Verwendung der zweiten kinematischen Gleichung

<img alt="Figure1" src="/files/ftp_upload/12623/12623_equation_1.png" style="text-align: center; height: 35px;" />

und Einsetzen der bekannten Größen in die obige Gleichung erhalten wir

<img alt="Figure2" src="/files/ftp_upload/12623/12623_equation_2.png" style="text-align: center; height: 35px;" />

<img alt="Figure3" src="/files/ftp_upload/12623/12623_equation_3.png" style="text-align: center; height: 15px;" />

Mit der quadratischen Formel ergeben sich zwei Lösungen für die Zeit: t = 10 s und t = -20 s. Ein negativer Wert für die Zeit ist unvernünftig, da das bedeuten würde, dass das Ereignis 20 s vor Beginn der Bewegung stattgefunden hat. Daher können wir diese Lösung verwerfen. Somit beträgt die Zeit, die das Auto benötigt, um die 200 m lange Auffahrt zu durchfahren, 10 s.

Dieser Text ist angepasst von <a href="https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration">Openstax, University Physics Volume 1, Abschnitt 3.4: Bewegung mit konstanter Beschleunigung.</a>

Kinematische Gleichungen - II

Die zweite kinematische Gleichung drückt die Endposition eines Objekts in Bezug auf seine Anfangsposition aus, die zurückgelegte Strecke mit der Anfangskonstanten ...

Education

Lehre

JoVE Core

Physik

Unit 1

Einheit 1

Motion Along a Straight Line

Bewegung entlang einer Geraden

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

using the second kinematic equation

The second kinematic equation relates an object’s final position to its initial position, its starting velocity, and the extra distance gained as velocity changes. It is used only when acceleration stays constant during the motion. That makes it useful for straight-line motion problems with a steady rate of acceleration.

A car merging onto a 200 m freeway ramp is a good example. In the example, the car starts at 10 m/s and accelerates at 2 m/s². The known values are the distance, the initial velocity, and the acceleration, so the second kinematic equation can be used to find the time.

Substituting the values into the equation leads to a quadratic equation for time. Solving it gives two answers: t = 10 s and t = −20 s. The negative time is not reasonable, because it would mean the motion happened 20 s before it started, so that solution is discarded. The time for the car to travel the 200 m ramp is 10 s.

Using the Second Kinematic Equation

position and displacement

Reference Frames and Motion in a Line

Position in straight-line motion depends on a frame of reference. A frame of reference is an arbitrary set of axes used to describe an object’s position and motion. It gives meaning to where something is located at a particular time. Earth is often used as the reference frame. In that case, we describe an object’s position relative to stationary objects on Earth. A rocket launch can be described this way. A cyclist’s position can also be described relative to the buildings they pass. <br...

average velocity

Average Velocity and Average Speed

Average velocity and average speed help describe motion in kinematics. To compare motion over time, we use the time variable. Time lets us track an object’s position and how fast that position changes. For motion between two points, each position x_i is paired with a time t_i. When the details of motion at each instant are not important, we use average velocity. Average velocity is a vector quantity, so it includes direction. It is found by dividing total displacement, Δx, by the time...

instantaneous velocity - i

Instantaneous Speed and Direction

Instantaneous velocity describes how fast an object is moving at a specific instant. It gives the motion of a particle at one point in time or at one point along its path. Average velocity over a time interval cannot show this level of detail. Instantaneous velocity, often written as v x , is the limit of average velocity as the elapsed time approaches zero. It can also be found as the derivative of x with respect to t . Like average velocity, it is a vector with dimensions of length per unit...

instantaneous velocity - ii

Instantaneous Velocity from Position Graphs

Instantaneous velocity describes how fast an object is moving at a specific moment along its path. It is the limit of average velocity as the elapsed time approaches zero. It can also be written as the derivative of displacement with respect to time. Like average velocity, instantaneous velocity is a vector, so it has both size and direction. Its units are length per unit time. Instantaneous velocity can be positive or negative. A position-time graph can show it at any point on the motion...

average acceleration

Average Acceleration and Velocity Change

Average acceleration describes how velocity changes over time. In physics, acceleration is not just about speeding up. It also includes slowing down and changing direction. Since velocity is a vector, it has both magnitude and direction. A change in velocity can come from a change in speed, a change in direction, or both. For example, a runner may travel at 10 km/h due east, stop, reverse direction, and then continue at 10 km/h due west. Even though the speed is the same, the velocity has...

instantaneous acceleration

Negative Acceleration and Instant Speed Change

Acceleration shows how velocity changes, and it does not always point in the same direction as motion. When an object slows down, its acceleration points opposite to its motion. This is often called deceleration, but that term can cause confusion because deceleration is not a vector and does not point in a specific direction in a coordinate system. A subway train is a simple example. When the train slows down, it is still accelerating, but the acceleration points opposite to the direction it...

kinematic equations - i

Velocity from Constant Acceleration

Velocity from constant acceleration shows how an object’s speed and direction change when the acceleration stays the same. The first kinematic equation connects initial velocity, acceleration, and time for motion with constant acceleration. It helps describe how velocity changes at a steady rate throughout the motion. The equation matches common sense in several cases. If acceleration is zero, the final velocity is the same as the initial velocity, so the object keeps a constant velocity. If...

kinematic equations - ii

The second kinematic equation relates an object’s final position to its initial position, its starting velocity, and the extra distance gained as velocity changes. It is used only when acceleration stays constant during the motion. That makes it useful for straight-line motion problems with a steady rate of acceleration. A car merging onto a 200 m freeway ramp is a good example. In the example, the car starts at 10 m/s and accelerates at 2 m/s². The known values are the distance, the initial...

kinematic equations - iii

Constant-Acceleration Kinematics Formulas

Constant-acceleration kinematics formulas connect an object’s motion when acceleration stays the same. They help describe how velocity and position change over time. These equations are useful for solving motion problems and for understanding the relationships among distance, time, velocity, and acceleration. The third kinematic equation does not use time as a variable. Instead, it gives final velocity in terms of acceleration and the distance over which that acceleration acts. The fourth...

kinematic equations: problem solving

Solving Constant Acceleration Motion Problems

Solving one-dimensional motion problems with constant acceleration starts with identifying the known quantities and choosing the right kinematic equation. A kinematic equation is a motion formula that relates displacement, velocity, acceleration, and time. Depending on what is known and unknown, you may need one equation or two equations to find the answer. In many problems, the number of equations needed matches the number of unknowns. Two-body pursuit problems are different. They always...

free-falling bodies: introduction

Gravity and Free-Fall Motion

Gravity and free-fall motion explain how objects move when only Earth’s gravity acts on them. In free-fall, the motion is caused by gravitational force, and air resistance is neglected. This means the object accelerates toward Earth’s center. This result is surprising because everyday experience includes friction and air resistance. We often expect lighter objects to fall more slowly than heavier ones. People held that idea until Galileo Galilei proved that mass does not change the...

free-falling bodies: example

Gravity-Driven Free Fall Motion

Gravity-driven free fall motion is the motion of an object when gravity is the only force acting on it and air resistance is absent. In this case, the object is in free-fall. The acceleration due to gravity stays constant, no matter how much the object weighs. Free-fall is not limited to objects moving straight down. It also includes motion that is affected only by gravity. Because the acceleration is constant, the motion can be described with kinematic equations. Objects dropped from the...

velocity and position by graphical method

velocity-and-position-by-graphical-method

Acceleration Graphs for Motion Prediction

Acceleration graphs can be used to find velocity and position over time. The area under an acceleration-time graph gives the change in velocity. The area under a velocity-time graph gives the change in position, or displacement. A real example is an airplane using an inertial navigation system. Before takeoff, the pilot enters the airplane’s initial position and velocity. The system then uses acceleration data to calculate the airplane’s velocity and position during flight. When acceleration...

velocity and position by integral method

Finding Motion from Acceleration Data

Velocity and position can be found from acceleration using integral calculus. When acceleration is known as a function of time, the velocity function comes from integrating acceleration. The position function then comes from integrating velocity. For constant acceleration, these integral steps match the first and second kinematic equations for velocity and position. The example in this lesson uses a motorboat that is moving at 5.0 m/s and then begins to slow down as it approaches the dock. Its...

key terms and concepts

Use the second kinematic equation to solve constant-acceleration motion problems. Follow a car on a 200 m ramp and find the travel time step by step.

Using the Second Kinematic Equation | JoVE K12

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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.

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Ohio’s Learning Standards and Model Curriculum

Motion

Problem Solving

Tennessee Academic Standards

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Motion and Stability: Forces and Interactions

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PS.8

PS.8.a

PH.2

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Florida Department of Education

SC.912.P.12

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Texas Essential Knowledge and Skills

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SP1.

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PS.Phy.2.2

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

Using the Second Kinematic Equation

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