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(2)=4quad" and "quad v(2)=6
Which of the following expression gives the velocity of the particle when 
t=9?

6+int_(9)^(2)a(t)dt

6+int_(9)^(2)v(t)dt

6+int_(2)^(9)a(t)dt

6+int_(2)^(9)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(2)=4 and v(2)=6 x(2)=4 \quad \text { and } \quad v(2)=6 \newlineWhich of the following expression gives the velocity of the particle when t=9? t=9 ? \newline6+92a(t)dt 6+\int_{9}^{2} a(t) d t \newline6+92v(t)dt 6+\int_{9}^{2} v(t) d t \newline6+29a(t)dt 6+\int_{2}^{9} a(t) d t \newline6+29v(t)dt 6+\int_{2}^{9} 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(2)=4 and v(2)=6 x(2)=4 \quad \text { and } \quad v(2)=6 \newlineWhich of the following expression gives the velocity of the particle when t=9? t=9 ? \newline6+92a(t)dt 6+\int_{9}^{2} a(t) d t \newline6+92v(t)dt 6+\int_{9}^{2} v(t) d t \newline6+29a(t)dt 6+\int_{2}^{9} a(t) d t \newline6+29v(t)dt 6+\int_{2}^{9} v(t) d t
  1. Understand Problem: To find the velocity of the particle at t=9t=9, we need to use the information given about the particle's velocity at t=2t=2 and how the velocity changes over time, which is given by the acceleration a(t)a(t).
  2. Calculate Initial Velocity: We know the velocity at t=2t=2 is v(2)=6v(2)=6. To find the velocity at t=9t=9, we need to add the change in velocity from t=2t=2 to t=9t=9 to the initial velocity at t=2t=2. The change in velocity is given by the integral of the acceleration function a(t)a(t) from t=2t=2 to t=9t=9.
  3. Determine Change in Velocity: The correct expression for the velocity at t=9t=9 is the initial velocity plus the integral of the acceleration from t=2t=2 to t=9t=9. This is represented by the expression:\newlinev(9)=v(2)+t=2t=9a(t)dtv(9) = v(2) + \int_{t=2}^{t=9} a(t) \, dt
  4. Calculate Final Velocity: Therefore, the correct expression from the given options is: 6+t=2t=9a(t)dt6 + \int_{t=2}^{t=9} a(t) \, dt
  5. Verify Correctness: The other options are incorrect because they either have the wrong limits of integration or are integrating the wrong function (velocity instead of acceleration).

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