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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)=7quad" and "quad v(0)=8
Which of the following expressions gives the distance traveled by the particle over the interval 
0 <= t <= 10 ?

int_(0)^(10)x(t)dt

int_(0)^(10)|v(t)|dt

int_(0)^(10)v(t)dt

int_(0)^(10)|x(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)=7 and v(0)=8 x(0)=7 \quad \text { and } \quad v(0)=8 \newlineWhich of the following expressions gives the distance traveled by the particle over the interval 0t10 0 \leq t \leq 10 ?\newline010x(t)dt \int_{0}^{10} x(t) d t \newline010v(t)dt \int_{0}^{10}|v(t)| d t \newline010v(t)dt \int_{0}^{10} v(t) d t \newline010x(t)dt \int_{0}^{10}|x(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)=7 and v(0)=8 x(0)=7 \quad \text { and } \quad v(0)=8 \newlineWhich of the following expressions gives the distance traveled by the particle over the interval 0t10 0 \leq t \leq 10 ?\newline010x(t)dt \int_{0}^{10} x(t) d t \newline010v(t)dt \int_{0}^{10}|v(t)| d t \newline010v(t)dt \int_{0}^{10} v(t) d t \newline010x(t)dt \int_{0}^{10}|x(t)| d t
  1. Introduction: To find the distance traveled by a particle, we need to consider the velocity of the particle and the time interval over which we are measuring. The distance traveled is the integral of the velocity function with respect to time, taking the absolute value if the velocity changes sign (indicating a change in direction). Let's examine the given options.
  2. Option 11: The first option, 010x(t)dt\int_{0}^{10}x(t)dt, represents the integral of the position function, which does not give us the distance traveled but rather the net change in position. This is not the correct expression for distance traveled.
  3. Option 22: The second option, 010v(t)dt\int_{0}^{10}|v(t)|\,dt, represents the integral of the absolute value of the velocity function. This is the correct expression for distance traveled, as it accounts for the total length of the path traveled without considering the direction of motion.
  4. Option 33: The third option, 010v(t)dt\int_{0}^{10}v(t)dt, represents the integral of the velocity function. This would give us the displacement, which is the net change in position, not the total distance traveled. If the velocity changes sign, this expression would not account for the distance traveled in the opposite direction.
  5. Option 44: The fourth option, 010x(t)dt\int_{0}^{10}|x(t)|dt, represents the integral of the absolute value of the position function, which does not have a clear physical interpretation in the context of distance traveled. This is not the correct expression for distance traveled.

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