4.3
Beschleunigungsvektoren
Im Alltag bedeutet Beschleunigen, dass man schneller wird. Die Beschleunigung ist ein Vektor in derselben Richtung wie die Änderung der Durchschnittsg…
Die Geschwindigkeit ist eine Vektorgröße und ist die Geschwindigkeit mit einer Richtung. Beschleunigung liegt vor, wenn es eine Änderung der Geschwindigkeit gibt – entweder durch Beschleunigen oder Verlangsamen – oder durch Richtungsänderung oder beides gleichzeitig.
Die mathematische Art, die Beschleunigung auszudrücken, ist die Änderung der Geschwindigkeit über die Zeit.
Angenommen, ein Objekt bewegt sich mit konstanter Geschwindigkeit. Beim Kurvenfahren wird es langsamer. Seine momentane Geschwindigkeit ändert sich sowohl in der Größe als auch in der Richtung. Seien die momentanen Geschwindigkeitsvektoren, die sich bei P1 und P2 befinden, die Vektoren v1 und v2.
Durch Subtrahieren der Vektoren erhält man die Geschwindigkeitsänderung, den Delta-Vektor v.
Nun liegt die Richtung des durchschnittlichen Beschleunigungsvektors in Richtung der Geschwindigkeitsänderung. Der mittlere Beschleunigungsvektor ist gegeben durch das Verhältnis zwischen dem Δ-Vektor v und Δt.
Wenn wir kleinere Zeitintervalle betrachten, nähert sich die Position P2 P1. In diesem Augenblick, wenn sich Δt Null nähert, nähert sich die durchschnittliche Beschleunigung der momentanen Beschleunigung.
View the full transcript and gain access to JoVE Core videos
Q1: What is the difference between speed and acceleration?
Speed is how fast an object moves, while acceleration is the rate of change in velocity. Acceleration occurs when velocity changes in magnitude, direction, or both. A runner traveling at constant speed but changing direction is accelerating, even though their speed remains the same. Acceleration is a vector quantity pointing in the direction of the velocity change.
Q2: How do you calculate average acceleration?
Average acceleration is calculated as the change in velocity divided by the change in time: a = Δv/Δt. The change in velocity is found by subtracting the initial velocity vector from the final velocity vector. The direction of average acceleration points in the same direction as the change in velocity. This ratio gives the rate at which velocity changes over a time interval.
Q3: What is instantaneous acceleration and how does it differ from average acceleration?
Instantaneous acceleration is the acceleration at a specific moment in time, found when the time interval approaches zero. As Δt becomes infinitesimally small, average acceleration approaches instantaneous acceleration. Instantaneous acceleration can be obtained from the derivative of the velocity function with respect to time. Average acceleration describes change over a finite interval, while instantaneous acceleration describes change at a precise instant.
Q4: Can an object accelerate while moving at constant speed?
Yes. An object accelerates whenever its velocity changes in direction, even if speed remains constant. A car traveling at constant speed around a curve is accelerating because its direction changes. Since velocity is a vector with both magnitude and direction, a change in either component constitutes acceleration. Direction change alone is sufficient to produce acceleration.
Q5: How does the direction of acceleration vectors relate to velocity change?
The direction of acceleration vectors always aligns with the direction of velocity change. When you subtract initial velocity from final velocity vectors, the resulting change in velocity determines the acceleration direction. This relationship holds whether the object speeds up, slows down, or changes direction. The acceleration vector points toward the direction the velocity is changing.
Q6: Why does a drag racer's acceleration change during a race?
A drag racer experiences large acceleration immediately after starting but acceleration tapers off as the vehicle approaches constant velocity. Acceleration varies significantly with time during motion because the rate of velocity change decreases. Once the racer reaches constant velocity, acceleration becomes zero. Average acceleration over the entire race differs from instantaneous acceleration at any particular moment.
Q7: Does an object's mass or size determine its acceleration?
No. Acceleration has nothing to do with an object's size or mass. Two objects of different masses can have identical accelerations if their velocities change at the same rate. Acceleration depends solely on how quickly velocity changes over time, not on the physical properties of the object. This independence makes acceleration a fundamental kinematic quantity independent of object characteristics.