If a Tesla Model S P100D in 'Ludicrous mode' is pushed to its limit, the first 3.0 s3.0\text{ s} of acceleration can be modeled as ax={(35 m/s3)t0 s≤t≤0.40 s14.6 m/s2−(1.5 m/s3)t0.40 s≤t≤3.0 sa_{x}=\begin{cases}(35\,\text{m/s}^3)t & 0\text{ s}\le t\le0.40\,\text{s}\\ 14.6\,\text{m/s}^2-(1.5\,\text{m/s}^3)t & 0.40\,\text{s}\le t\le3.0\,\text{s}\end{cases} What acceleration would be needed to achieve the same speed in the same time at constant acceleration? Give your answer as a multiple of gg.

Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

Problem 69b

Chapter 2, Problem 69b

If a Tesla Model S P100D in 'Ludicrous mode' is pushed to its limit, the first $3.0 s$ of acceleration can be modeled as
$a_{x} = {\begin{matrix} (35 {m/s}^{3}) t & 0 s \leq t \leq 0.40 s \\ 14.6 {m/s}^{2} - (1.5 {m/s}^{3}) t & 0.40 s \leq t \leq 3.0 s \end{matrix}$
What acceleration would be needed to achieve the same speed in the same time at constant acceleration? Give your answer as a multiple of $g$ .

Verified step by step guidance

Step 1: Understand the problem. The Tesla's acceleration is given as a piecewise function of time. To find the constant acceleration that achieves the same final speed in the same time, we first need to calculate the total change in velocity (Δv) over the 3.0 s using the given acceleration function.

Step 2: Break the problem into two time intervals based on the piecewise function. For the first interval (0 ≤ t ≤ 0.40 s), the acceleration is a_x = (35 m/s³)t. Use the formula for velocity change, Δv = ∫a_x dt, to integrate this acceleration over the interval.

Step 3: For the second interval (0.40 s ≤ t ≤ 3.0 s), the acceleration is a_x = 14.6 m/s² - (1.5 m/s³)t. Again, use the formula Δv = ∫a_x dt to integrate this acceleration over the interval. Add the results from both intervals to find the total change in velocity (Δv).

Step 4: Once the total change in velocity (Δv) is determined, use the kinematic equation for constant acceleration: v_f = a_c * t, where v_f is the final velocity, a_c is the constant acceleration, and t is the total time (3.0 s). Solve for a_c = Δv / t.

Step 5: To express the constant acceleration as a multiple of g (gravitational acceleration, approximately 9.8 m/s²), divide the calculated constant acceleration (a_c) by g: a_c / g. This gives the acceleration in terms of g.

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Key Concepts

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

Acceleration

Acceleration is the rate of change of velocity of an object with respect to time. It is a vector quantity, meaning it has both magnitude and direction. In the context of the Tesla Model S, the acceleration can vary over time, as indicated by the piecewise function provided in the question. Understanding how to calculate and interpret acceleration is crucial for determining the forces acting on the vehicle and its performance.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this question, kinematic equations can be used to analyze the motion of the Tesla Model S under different acceleration conditions, allowing for the calculation of speed over time. This understanding is essential for solving the problem of achieving a specific speed under constant acceleration.

Gravitational Acceleration (g)

Gravitational acceleration, denoted as 'g', is the acceleration due to Earth's gravity, approximately 9.81 m/s². It serves as a reference point for measuring other accelerations. In this question, expressing the required acceleration as a multiple of 'g' allows for a clearer comparison of the Tesla's performance to a familiar standard. This concept is important for understanding how the vehicle's acceleration relates to everyday experiences of gravity.

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