Help with trigonmetric function question

[maccaman]

This is a problem solving one, i just don't know where to start it though

The speed of a particle in an alternating current is given by the equation:

v = 3 cos t/3, where v is in cm/s and t is in seconds.

The particle's movement begins at its point of equilibrium.
How soon after the start would the particle first reach a displacement of
-4.5cm?

 

[Integral]

You are given an expression for the velocity, you need to find the displacement.

$$v = \frac {dx} {dt}$$

so

$$\frac {dx} {dt} = 3 cos (\frac t 3)$$

$$\int dx = \int 3 cos (\frac t 3) dt$$

You should now be able to find an expression for the displacement as a function of time. You will need to find the time that satifies your conditions.

 

[M98Ranger]

To solve this problem, we can use the trigonometric identity cos(t) = -cos(t + π). This identity allows us to find the time at which the particle reaches a displacement of -4.5cm by setting the equation equal to -4.5 and solving for t.

-4.5 = 3 cos t/3

cos t/3 = -1.5

Using the inverse cosine function, we can find the value of t that satisfies this equation.

t/3 = cos^-1(-1.5)

t = 3 cos^-1(-1.5)

Now, using a calculator, we can find that cos^-1(-1.5) is approximately 2.618 radians.

Therefore, t = 3(2.618) = 7.854 seconds.

Thus, the particle would first reach a displacement of -4.5cm after 7.854 seconds from the start of its movement.