A car's velocity as a function of time is given by v x ( t ) = α + β t 2 v_x(t) = α + βt^2 , where α = 3.00 α = 3.00 m/s and β = 0.100 β = 0.100 m/s 3 . Calculate the average acceleration for the time interval t = 0 t = 0 to t = 5.00 t = 5.00 s.

Welcome back everybody. We are making observations about a squirrel that is running along the ground around the forest. We are given the velocity of the squirrel as a function of time. According to this function right here, Alpha, which is equal to 4.5 plus beta which were given as 0.27 times t squared. And we are looking at an interval where the starting time is zero seconds and the final time is 12 seconds. And we are tasked with finding what is the squirrels. Average acceleration over this time? Well, we have a function for this. This is just going to be equal to the change in velocity over the change in time which is equal to our final velocity minus our initial velocity divided by our final time minus our initial time. What we have these bottom terms down here, But how are we going to find these top terms up here? What we do is we just plug in these times into our formula for velocity to find our initial and final velocities. So let's go ahead and do that first. I want to start out with our initial velocity. This is just going to be equal to us. Plugging in zero into our equation. This gives us 4.5 plus 0.27 times zero squared. This whole term goes away and this gives us 4.5 m per second. Great! Now what about our final velocity? Well, all we do is just plug in our final time of 12 seconds and we get 4.5 plus 0.27 times 12 squared. Which one? We plug into our calculator. We get 43.38 m per second. Now we are ready to use our equation right here to find the average acceleration. So we have that our average acceleration is equal to 43. -4.5, divided by 12 0, giving us a final answer choice of 2.865 m/s squared or answer choice. C Thank you all so much for watching. Hope this video helped. We will see you all in the next one.

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2. 1D Motion / Kinematics

Intro to Acceleration

Physics

2. 1D Motion / Kinematics

Intro to Acceleration

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2:34 minutes

Problem 17a

Textbook Question

A car's velocity as a function of time is given by $v_{x} (t) = α + β t^{2}$ , where $alpha equals 3.00$ m/s and $beta equals 0.100$ m/s³. Calculate the average acceleration for the time interval $t equals 0$ to $t equals 5.00$ s.

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Understand the problem: We need to calculate the average acceleration of a car over a given time interval using the velocity function v_x(t) = α + βt^2, where α = 3.00 m/s and β = 0.100 m/s^3.

Recall the formula for average acceleration: Average acceleration is defined as the change in velocity divided by the change in time. Mathematically, it is expressed as:

\frac{Δ v}{Δ t}

Calculate the initial and final velocities: Use the given velocity function v_x(t) = α + βt^2 to find the initial velocity at t = 0 and the final velocity at t = 5.00 s. Substitute t = 0 and t = 5.00 s into the equation to find v_x(0) and v_x(5).

Determine the change in velocity: Calculate the change in velocity by subtracting the initial velocity from the final velocity:

Δ v = v_x(5) - v_x(0)

Calculate the average acceleration: Use the formula for average acceleration with the calculated change in velocity and the time interval (5.00 s - 0 s) to find the average acceleration:

\frac{Δ v}{Δ t}

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Velocity as a Function of Time

Velocity as a function of time describes how an object's speed and direction change over time. In this problem, the velocity is given by the equation v_x(t) = α + βt^2, where α is the initial velocity and βt^2 represents the change in velocity over time due to acceleration. Understanding this equation is crucial for determining how velocity evolves during the specified time interval.

Average Acceleration

Average acceleration is defined as the change in velocity divided by the time over which the change occurs. It provides a measure of how quickly an object's velocity changes on average over a given time period. In this problem, calculating the average acceleration involves finding the difference in velocity at the start and end of the time interval and dividing by the duration of the interval.

Integration of Velocity Function

To find the change in velocity over a time interval when given a velocity function, one can integrate the function with respect to time. In this context, integrating the velocity function v_x(t) = α + βt^2 from t = 0 to t = 5.00 s will yield the total change in velocity, which is necessary for calculating the average acceleration. This process involves applying basic calculus principles to solve the problem.

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Textbook Question

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A good model for the acceleration of a car trying to reach top speed in the least amount of time is a_𝓍 = a_₀ ─ kv_𝓍, where a_₀ is the initial acceleration and k is a constant. Find an expression for the car's velocity as a function of time.

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A car's velocity as a function of time is given byvx(t)=α+βt2 v_x(t) = α + βt^2, where α=3.00α = 3.00 m/s and β=0.100β = 0.100 m/s3. Calculate the average acceleration for the time interval t=0t = 0 to t=5.00t = 5.00 s.

Key Concepts

Velocity as a Function of Time

Average Acceleration

Integration of Velocity Function

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A car's velocity as a function of time is given by $v_{x} (t) = α + β t^{2}$ , where $alpha equals 3.00$ m/s and $beta equals 0.100$ m/s³. Calculate the average acceleration for the time interval $t equals 0$ to $t equals 5.00$ s.