a particle moves in a straight line such that its position x from a fixed point 0 at time 't' is given by

x= 5 + 8sin2t + 6cos2t

1. Find the period and amplitude of the particle.

2. Find the greatest speed of the particle.

Could you please explain the steps on how to get to the answer?
thanks!

As you have surely learned,

a cos kx + b sin kx has
period 2pi/k
amplitude √(a^2+b^2)

plug and chug.

Certainly! To find the period and amplitude of the particle's motion, we need to analyze the equation.

1. Find the period and amplitude:
The general equation for a simple harmonic motion is given by x = A*sin(ωt + φ), where A is the amplitude, ω is the angular frequency, t is the time, and φ is the phase angle.

In our case, x = 5 + 8sin(2t) + 6cos(2t). We can see that there are two components in the equation, sin(2t) and cos(2t), which means the motion is a combination of both sine and cosine functions.

To find the period, we need to identify the component that repeats itself. From the given equation, we observe that sine and cosine functions have the same argument, 2t. This means the period will be the same for both functions.

The period of a sine or cosine function is given by T = 2π/ω, where ω is the angular frequency. In our case, the angular frequency is 2 because of the 2t argument. Therefore, the period T = 2π/2 = π.

Hence, the period of the particle's motion is π.

To find the amplitude, we can use the equation A = sqrt(A_sin^2 + A_cos^2), where A_sin is the coefficient of sin(2t) and A_cos is the coefficient of cos(2t). In our equation, A_sin = 8 and A_cos = 6, so the amplitude A = sqrt(8^2 + 6^2) = 10.

So, the amplitude of the particle's motion is 10.

2. Find the greatest speed of the particle:
To find the greatest speed, we can differentiate the equation with respect to time (t).

Differentiating x = 5 + 8sin(2t) + 6cos(2t) with respect to t, we get:

dx/dt = 2*8cos(2t) - 2*6sin(2t)
= 16cos(2t) - 12sin(2t)

Now, to find the maximum value of dx/dt (the greatest speed), we need to find the conditions where the derivative is equal to zero, i.e., dx/dt = 0.

So, 16cos(2t) - 12sin(2t) = 0

Rearranging the equation, we get:

4cos(2t) = 3sin(2t)

Dividing both sides by cos(2t), we get:

tan(2t) = 4/3

Now, we can find the values of 2t using the inverse tangent function.

2t = atan(4/3)

Since tangent is positive in the first and third quadrants, we focus on the first quadrant values only.

t = atan(4/3) / 2

Using a calculator, we find t ≈ 0.588 rad.

Finally, to find the greatest speed, we substitute the value of t into the derivative: dx/dt = 16cos(2t) - 12sin(2t).

So, the greatest speed of the particle occurs at t ≈ 0.588 rad.

I hope that helps! Let me know if you have any more questions.