Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

A particle moves along the 
x-axis such that at any time 
t >= 0 its position is 
x(t), its velocity is 
v(t), and its acceleration is 
a(t).
What is the average acceleration of the particle on the interval 
0 <= t <= 8?

(x(8)-x(0))/(8)

(a(8)-a(0))/(8)

(1)/(8)int_(0)^(8)a(t)dt

(1)/(8)int_(0)^(8)v(t)dt

A particle moves along the x x -axis such that at any time t0 t \geq 0 its position is x(t) x(t) , its velocity is v(t) v(t) , and its acceleration is a(t) a(t) .\newlineWhat is the average acceleration of the particle on the interval 0t8? 0 \leq t \leq 8 ? \newlinex(8)x(0)8 \frac{x(8)-x(0)}{8} \newlinea(8)a(0)8 \frac{a(8)-a(0)}{8} \newline1808a(t)dt \frac{1}{8} \int_{0}^{8} a(t) d t \newline1808v(t)dt \frac{1}{8} \int_{0}^{8} v(t) d t

Full solution

Q. A particle moves along the x x -axis such that at any time t0 t \geq 0 its position is x(t) x(t) , its velocity is v(t) v(t) , and its acceleration is a(t) a(t) .\newlineWhat is the average acceleration of the particle on the interval 0t8? 0 \leq t \leq 8 ? \newlinex(8)x(0)8 \frac{x(8)-x(0)}{8} \newlinea(8)a(0)8 \frac{a(8)-a(0)}{8} \newline1808a(t)dt \frac{1}{8} \int_{0}^{8} a(t) d t \newline1808v(t)dt \frac{1}{8} \int_{0}^{8} v(t) d t
  1. Understand average acceleration: Understand the concept of average acceleration.\newlineAverage acceleration is defined as the change in velocity (Δv\Delta v) over the change in time (Δt\Delta t). Since acceleration is the derivative of velocity with respect to time, the average acceleration over an interval can be found by integrating the acceleration function over that interval and then dividing by the length of the interval (1Δta(t)dt\frac{1}{\Delta t}\int a(t) dt).
  2. Identify correct formula: Identify the correct formula for average acceleration.\newlineThe average acceleration over the interval from t=0t = 0 to t=8t = 8 is given by the integral of the acceleration function a(t)a(t) from 00 to 88, divided by the length of the interval, which is 88. This is represented by the formula:\newlineAverage acceleration = (1/8)×08a(t)dt(1/8) \times \int_{0}^{8} a(t) \, dt
  3. Apply formula to calculate: Apply the formula to calculate the average acceleration.\newlineWe are given the expression for average acceleration as 18\frac{1}{8} * 08a(t)dt\int_{0}^{8} a(t) \, dt. This is the correct expression to use, and we do not need to perform any further calculations since we are not given a specific function for a(t)a(t).

More problems from Find derivatives of sine and cosine functions