Kinematics: Acceleration Vs Time Graph

Summary

This educational video script explores the concept of acceleration versus time, illustrating how the area under the graph represents change in velocity. It explains the relationship between acceleration, time, and velocity using mathematical formulas. The script also includes a practical example of a tricycle's motion, calculating its final velocity after 2.0 seconds given an acceleration function. The video aims to clarify the physical principles behind acceleration and velocity changes, making complex physics concepts more accessible.

Takeaways

• 📈 The script discusses the concept of acceleration versus time graphs, explaining how the area under the graph represents change in velocity.
• ⏱️ It introduces the term 'delta t' or change in time, which is a measure of time intervals on the x-axis of the graph.
• 🔢 The area under the graph is likened to the formula for volume, where height times width gives the area, and in this case, average acceleration times change in time gives the change in velocity.
• 🌟 The formula for change in velocity (delta v) is given as 'a times delta t', which is the area under the acceleration-time graph.
• 📚 The script explains the mathematical limit definition of change in velocity as delta t approaches zero, which is the summation of 'a sub i times delta t sub i' from i=1 to n.
• 🧮 It provides a detailed example of calculating final velocity by integrating the acceleration function from the initial time to the final time.
• 🚴‍♂️ The example involves a tricycle rider named Circus who starts from a position 5.0 meters away from the origin with an acceleration function of '10.0 m/s^2 + 3.0 m/s^4 * t^2'.
• ⏳ The calculation for the rider's velocity after 2.0 seconds is demonstrated, involving the integration of the acceleration function over the time interval.
• 🔍 The anti-derivatives of the acceleration components are calculated, and the limits of integration from 0 to 2.0 seconds are applied to find the velocity.
• 🏁 The final velocity is computed to be 28 meters per second after applying the limits of integration and simplifying the expression.

Q & A

What is the relationship between acceleration and the area under the acceleration versus time graph?

The area under the acceleration versus time graph represents the change in velocity (Δv), which can be calculated as the integral of acceleration over time.

What does the term 'delta t' signify in the context of the script?

In the script, 'delta t' refers to the change in time (Δt), which is the difference between the final time and the initial time.

How is the final velocity of an object calculated when given its acceleration as a function of time?

The final velocity (v) is calculated by adding the initial velocity (v₀) to the integral of acceleration (a(t)) with respect to time from the initial time (t₀) to the final time (t).

What is the formula for calculating the change in velocity when acceleration is constant?

When acceleration is constant, the change in velocity (Δv) is given by the formula Δv = a * Δt, where 'a' is the constant acceleration and 'Δt' is the change in time.

What does the term 'a sub i' represent in the script?

The term 'a sub i' likely represents the acceleration at a specific time interval 'i' when acceleration is not constant and varies over time.

How is the integral of acceleration over time related to the change in velocity?

The integral of acceleration over time is equal to the change in velocity, as acceleration is the rate of change of velocity.

What is the significance of the integral from t₀ to t of 'a of t' with respect to time in the script?

This integral represents the total change in velocity over the time interval from t₀ to t, given the acceleration as a function of time.

In the example provided, what is the initial displacement of the tricycle?

The initial displacement of the tricycle is 5.0 meters from its original position.

What is the expression for the acceleration of the tricycle in the example, and how does it change over time?

The acceleration of the tricycle is given by the expression 10.0 m/s² + 3.0 m/s⁴ * t², which means the acceleration increases quadratically over time.

What is the final velocity of the tricycle after 2.0 seconds, as calculated in the script?

The final velocity of the tricycle after 2.0 seconds is 28.0 meters per second, calculated by integrating the acceleration function over time and adding it to the initial velocity.

Outlines

00:00

📚 Introduction to Calculating Velocity from Acceleration

This paragraph introduces the concept of calculating velocity from acceleration using the area under the acceleration versus time graph. It explains that acceleration is the change in velocity over time, and the area under the graph represents this change. The formula for change in velocity (delta v) is given by the integral of acceleration (a) over time (t), which is also represented as the limit of the sum of products of acceleration and small time intervals as these intervals approach zero. The paragraph uses the example of a tricycle's motion to illustrate the concept, where the tricycle's acceleration is given as a function of time, and the task is to find its velocity after a certain time.

05:01

🔍 Detailed Calculation of Velocity Using Integration

This paragraph delves into the detailed calculation of the tricycle's velocity after 2.0 seconds, as described in the previous paragraph. It outlines the process of integrating the acceleration function, which includes a constant term and a time-dependent term, over the given time interval from 0 to 2.0 seconds. The integration is performed step by step, with the anti-derivatives for both terms being calculated. The final velocity is obtained by evaluating these anti-derivatives at the upper and lower limits of the time interval and adding them together. The result shows the final velocity of the tricycle after 2.0 seconds, which is a sum of the contributions from the constant and time-dependent acceleration terms.

Keywords

💡Acceleration

Acceleration is the rate of change of velocity with respect to time. It is a vector quantity that describes how quickly an object is speeding up or slowing down. In the video, acceleration is a central concept, as it is used to calculate the change in velocity over time. The script mentions an acceleration of '10.0 meters per second squared plus 3.0 meter per second to the power of 4 times t squared,' which is a variable acceleration that changes over time, illustrating how acceleration can be constant or can vary depending on the situation.

💡Time

Time is a measure in which events can be ordered from the past through the present into the future. In the context of the video, time is used to quantify the duration over which acceleration occurs. The script refers to 'time equals zero' as the starting point and 'final time i 2.0 seconds' as the end point for calculating the change in velocity, highlighting the importance of time in kinematic equations.

💡Velocity

Velocity is a vector quantity that represents the rate of change of an object's position with respect to time. It includes both speed (a scalar quantity) and direction. The video script discusses calculating the final velocity of an object after a given time period by integrating the acceleration over that time. The example given is 'his velocity after 2.0 seconds,' which is calculated by integrating the acceleration function from the initial time to the final time.

💡Displacement

Displacement is the change in position of an object. It is a vector that represents the shortest path from the initial to the final position, regardless of the path taken. In the script, 'initial displacement at 5.0 meters' is mentioned, which sets the starting point for the calculation of the object's motion and its change in position over time.

💡Integral

An integral in calculus represents the sum of an infinite number of infinitesimally small quantities. In the video, integration is used to find the area under the acceleration versus time graph, which corresponds to the change in velocity. The script states 'integral from t sub o to t of a of t dt,' which is the mathematical operation to calculate the total change in velocity over the time interval from the initial to the final time.

💡Area Under the Curve

The area under a curve in a graph represents the accumulation of a quantity over a given interval. In the context of the video, the area under the acceleration versus time graph is used to find the total change in velocity. The script explains that 'the area under the curve of acceleration versus time graph is change in velocity,' which is a fundamental concept in physics for understanding how motion changes over time.

💡Anti-Derivative

An anti-derivative is a function whose derivative is equal to the original function. In the video, anti-derivatives are used to find the velocity by integrating the acceleration function. The script mentions 'anti-derivative 10.0 meter per second squared' and another for '3.0 meter per second raised to 4 times t squared,' which are used to calculate the velocity at a given time by reversing the process of differentiation.

💡Limit

In calculus, a limit is the value that a function or sequence approaches as the input approaches a certain value. The script refers to 'limit where delta t approaches zero' in the context of finding the instantaneous rate of change, which is a fundamental concept in understanding how calculus is used to analyze motion and acceleration.

💡Summation

Summation is the operation of adding a sequence of numbers to find their total. In the video, summation is mentioned in the context of adding up infinitesimally small changes in velocity to find the total change over a time interval. The script uses 'summation from i equal to one to n of a sub i times delta t sub i' to illustrate the process of summing up these small changes, which is a precursor to the integral in calculus.

💡Change in Velocity (Delta V)

Change in velocity, often denoted as delta V, is the difference between the final and initial velocities of an object. In the video, delta V is calculated by integrating the acceleration over time, which is a key step in determining how the velocity of an object changes. The script provides a formula 'delta v equals a delta t' and further explains it through the example of calculating the velocity after 2.0 seconds.

Highlights

Introduction of the concept of acceleration versus time graph

Explanation of the area under the graph representing change in velocity

Formula for calculating change in velocity using the area under the acceleration-time graph

Derivation of the formula for final velocity using integration

Example problem involving a tricycle with a variable acceleration

Calculation of the tricycle's velocity after 2.0 seconds using the derived formula

Use of anti-derivatives to solve for the change in velocity

Integration of the acceleration function to find the velocity

Explanation of the limits of integration in the context of the problem

Final calculation of the tricycle's velocity resulting in 28 meters per second

Emphasis on the importance of the area under the curve in physics problems

Illustration of how to apply the concept of integration to real-world physics scenarios

Clarification of the difference between average and instantaneous acceleration

Demonstration of how to calculate the area under a curve using integration

Application of the concept of limits in calculus to the physics problem

Detailed step-by-step solution to the example problem provided

Use of specific values to demonstrate the calculation process

Final conclusion of the example problem with a clear answer

Transcripts

[Music]

the acceleration versus time graph

and time i eat flat nothing

x-axis

[Music]

[Music]

[Music]

long and final time i am delta t or

change in time

an area under the graph i knock of koha

gambit and height times width

and volume and height i am average

acceleration

atomic volume width i am change in time

[Music]

panzernine area equals

a times delta t i am formula

parama and delta v or change in velocity

kayang area under the acceleration by

the time graph

ion change in velocity delta v

equals a delta t

[Music]

is equal to limit where delta t

approaches zero

of the summation from i equal to one

to n of a sub i times

delta t sub i at ito

i integral from t sub o to t

of a of t dp

an area under the curve ion change in

velocity

and final velocity ito i

the b sub equals plus integral from t

sub o to t

of a of t dt

gonna undo the function i though an

area under the curve i integral from t

sub o to t

of a of t dt at ito

i am delta v or change in velocity

the area under the curve of acceleration

versus time graph

i change in velocity

at and final velocity i b sub o

plus integral from t sub o to t of

a of t dt

[Music]

halimbawa circus rode his tricycle

along a straight road at time equals

zero

he started traveling and washed

5.0 meters away from its original

position

his acceleration was 10.0 meter per

second squared

plus 3.0 meter per second raised to 4

times t squared what is his velocity

after 2.0 seconds

alma given i initial time

i zero final time i 2.0 seconds

initial displacement at 5.0 meters

initial velocity i zero at an a of t

i ten point zero meter per second

squared

plus three point zero meter per second

raised to four

times t squared and hina

a final velocity and solution i

v equals v sub o

plus integral from t sub o to t of

a of t dt is a substitute nothing

i 10.0 meter per second squared plus 3.0

meter per second raised to four

and t sub o i zero at n

t i 2.0 seconds

an anti-derivative 10.0 meter per second

squared

i meter per second squared times

t at an anti-derivative

and 3.0 meter per second raised to 4

times t

squared i 3.0 meter per second raised to

4

times t rate three over three

and lower limit at zero at an upper

limit at 2.0 seconds

[Music]

velocity i 10.0 meter per second squared

times 2.0 seconds plus 3.0 meter per

second raised to 4

times 2.0 second raised to 3 over 3.

10.0 meter per second times 2.0

at the end at 20.0 meter per second

cancelled out din on three over three

at cancelled out then an s raise to

three

at s raised to four ang

i 8.0 meter per second

and twenty point zero meter per second

plus eight point zero meter per second

i 28 meter per second at the end and

final velocity

right now

you