Question

2. A particle moves along the x-axis so that any time t>_ 0, its velocity is given by v(t) = sin(2t). If the position of the particle at time t=pi/2 is x=4, what is the particle's position at time t=0?

Answer 1

To find the particle's position at time t = 0, you need to integrate the given velocity function and substitute the given position at t = pi/2 into the position function to find the value of the integration constant. The position at t = 0 can then be found by substituting t = 0 into the position function.

Step-by-step explanation:

To find the particle's position at time t = 0, we can integrate the given velocity function. The position function is found by integrating the velocity function with respect to time. In this case, the integral of sin(2t) is -cos(2t) + C, where C is a constant. Since the position at t = pi/2 is given as x = 4, we can substitute t = pi/2 and x = 4 into the position function to find the value of C. Hence, -cos(2(pi/2)) + C = 4, and -cos(pi) + C = 4. Therefore, C = 3. The position function is x(t) = -cos(2t) + 3. To find the position at t = 0, we substitute t = 0 into the position function. Hence, x(0) = -cos(2(0)) + 3 = -cos(0) + 3 = -1 + 3 = 2. Therefore, the particle's position at time t = 0 is x = 2.

Answer 2

Since a particle moves along the x-axis so that any time t ≥ 0, its velocity is given by v(t) = sin(2t), the particle's position at time t = 0 is 3.5

Since a particle moves along the x-axis so that any time t ≥ 0, its velocity is given by v(t) = sin(2t). If the position of the particle at time t=pi/2 is x = 4. To find what is the particle's position at time t = 0, we proceed as follows

Since we have the velocity of the particle as v(t) =sin(2t), to find its position function, we integrate its velocity function since v = dx/dt

dx = vdt

x = ∫vdt

So, substituting v into the equation, we have that

x = ∫vdt

x = ∫sin(2t)dt

x = -cos(2t)/2 + C

Now, at t = π/4, x = 4. So, substituting htese into the equation, we have that

x = -cos(2t)/2 + C

4 = -cos(2π/4)/2 + C

4 = -cos(π/2)/2 + C

4 = 0/2 + C

4 = 0 + C

C = 4

So, the position function is x(t) = -cos(2t)/2 + 4

Now at t = 0, we have that the position is

x(0) = -cos(2(0))/2 + 4

x(0) = -cos0/2 + 4

x(0) = -1/2 + 4

x(0) = (8 - 1)/2

x(0) = 7/2

x(0) = 3.5

So, the position of the particle at t = 0 is 3.5