How Do You Find the First Time Position and Velocity Reach Their Maximum Values?

[lefthand]

this here is the problem...

Use the equations below. (Note that the direction is indicated by the sign in front of the equations.)
x = (0.0553 m) cos(4.094t + 0.14π)

v = −(0.226 m/s) sin(4.094t + 0.14π)

a = −(0.927 m/s2) cos(4.094t + 0.14π)

(a) Determine the first time (t > 0) that the position is at its maximum (positive) value.
s

(b) Determine the first time (t > 0) that the velocity is at its maximum (positive) value.
s

i have no idea where to start, or how to do this.

 

[grzz]

Position is given by x.

At what angle is the cos(...) maximum?

 

[lefthand]

they didnt give an angle. but I'm guessing since we want x to equal what ever number is there, then cos must equal 1 right?

 

[grzz]

so (0.0553 m) cos(4.094t + 0.14π) is max when cos(4.094t + 0.14π) = 1

that is when (4.094t + 0.14π) = ...?

 

[asteriatic]

I would suggest breaking down the problem into smaller parts and utilizing the equations provided. First, let's focus on part (a) and finding the time at which the position is at its maximum value. To do this, we can set the equation for position (x) equal to zero and solve for t. This will give us the time at which the position is at its maximum (positive) value.

0 = (0.0553 m) cos(4.094t + 0.14π)

We can then simplify the equation by dividing both sides by (0.0553 m) and taking the inverse cosine of both sides.

cos^-1(0) = 4.094t + 0.14π

This gives us the equation:

t = (-0.14π)/4.094

Solving for t, we get t = -0.034 seconds. However, the problem states that t must be greater than 0, so we can discard this solution. We can also see from the given equations that the position will be at its maximum value every time cos(4.094t + 0.14π) equals 1. This occurs when the angle inside the cosine function is equal to zero. So, we can set 4.094t + 0.14π = 0 and solve for t to find the first time that the position is at its maximum value.

4.094t + 0.14π = 0

Solving for t, we get t = (-0.14π)/4.094 = -0.034 seconds.

Therefore, the first time (t > 0) that the position is at its maximum value is 0.034 seconds.

For part (b), we can approach it in a similar manner. The velocity will be at its maximum (positive) value every time sin(4.094t + 0.14π) equals 1. This occurs when the angle inside the sine function is equal to π/2. So, we can set 4.094t + 0.14π = π/2 and solve for t to find the first time that the velocity is at its maximum value.

4.094t + 0.14π = π/2

Solving for t, we get t = (π/2 - 0.14π)/4.094 = 0.