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). You are given:

x(0)=4quad" and "quad v(0)=6
Which of the following expressions gives the distance traveled by the particle over the interval 
0 <= t <= 7 ?

4+int_(0)^(7)|v(t)|dt

int_(0)^(7)|v(t)|dt

int_(0)^(7)v(t)dt

4+int_(0)^(7)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) . You are given:\newlinex(0)=4 and v(0)=6 x(0)=4 \quad \text { and } \quad v(0)=6 \newlineWhich of the following expressions gives the distance traveled by the particle over the interval 0t7 0 \leq t \leq 7 ?\newline4+07v(t)dt 4+\int_{0}^{7}|v(t)| d t \newline07v(t)dt \int_{0}^{7}|v(t)| d t \newline07v(t)dt \int_{0}^{7} v(t) d t \newline4+07v(t)dt 4+\int_{0}^{7} 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) . You are given:\newlinex(0)=4 and v(0)=6 x(0)=4 \quad \text { and } \quad v(0)=6 \newlineWhich of the following expressions gives the distance traveled by the particle over the interval 0t7 0 \leq t \leq 7 ?\newline4+07v(t)dt 4+\int_{0}^{7}|v(t)| d t \newline07v(t)dt \int_{0}^{7}|v(t)| d t \newline07v(t)dt \int_{0}^{7} v(t) d t \newline4+07v(t)dt 4+\int_{0}^{7} v(t) d t
  1. Initial Position and Change: To find the distance traveled by the particle, we need to consider the initial position and the change in position due to the velocity over the time interval. The initial position is given by x(0)=4x(0) = 4. The change in position is given by the integral of the velocity function v(t)v(t) over the time interval from 00 to 77. However, since we are interested in the distance and not the displacement, we need to take the absolute value of the velocity to ensure that we are adding up all the distances traveled, regardless of the direction.
  2. Absolute Value of Velocity: The correct expression for the distance traveled is the initial position plus the integral of the absolute value of the velocity function over the given time interval. This is because the initial position gives us the starting point, and the integral of the absolute value of the velocity gives us the total distance traveled without considering the direction.
  3. Expression for Distance Traveled: The expression that correctly represents the distance traveled by the particle from time t=0t = 0 to t=7t = 7 is therefore:\newline4+07v(t)dt4 + \int_{0}^{7} |v(t)| \, dt\newlineThis expression takes into account the initial position and adds the total distance traveled, as indicated by the absolute value of the velocity function integrated over the time interval.

More problems from Relate position, velocity, speed, and acceleration using derivatives