Productive Task 




A particle is moving along a horizontal line in simple harmonic motion. The displacement of the particle at time t is given by:

x(t) = 5 (sin (2t))5

Find the velocity [w(t) = x'(t)] of the

particle at time t =

π /4



To find the velocity of the particle at time \( t = \frac{\pi}{4} \), we first need to differentiate the displacement function \( x(t) = 5 (\sin(2t))^5 \) to find \( w(t) = x'(t) \).

### Step 1: Differentiate \( x(t) \)

Using the chain rule, we differentiate \( x(t) \):

\[
x(t) = 5 (\sin(2t))^5
\]

Let \( u = \sin(2t) \). Then, \( x(t) = 5u^5 \). Using the chain rule, we have:

\[
\frac{dx}{dt} = 5 \cdot 5u^4 \cdot \frac{du}{dt}
\]

Now, calculate \( \frac{du}{dt} \):

\[
\frac{du}{dt} = \cos(2t) \cdot 2 = 2\cos(2t)
\]

Now substituting back:

\[
\frac{dx}{dt} = 25u^4 \cdot (2\cos(2t)) = 50 (\sin(2t))^4 \cos(2t)
\]

Thus, the velocity function \( w(t) \) is:

\[
w(t) = 50 (\sin(2t))^4 \cos(2t)
\]

### Step 2: Evaluate at \( t = \frac{\pi}{4} \)

Now, substitute \( t = \frac{\pi}{4} \):

\[
w\left(\frac{\pi}{4}\right) = 50 \left(\sin\left(2 \cdot \frac{\pi}{4}\right)\right)^4 \cos\left(2 \cdot \frac{\pi}{4}\right)
\]

Calculating the sine and cosine:

\[
\sin\left(\frac{\pi}{2}\right) = 1, \quad \cos\left(\frac{\pi}{2}\right) = 0
\]

Thus,

\[
w\left(\frac{\pi}{4}\right) = 50 \cdot (1)^4 \cdot 0 = 0
\]

### Conclusion

The velocity of the particle at \( t = \frac{\pi}{4} \) is:

\[
w\left(\frac{\pi}{4}\right) = 0
\]

If you have any questions or need further details, feel free to ask! Here are some related questions you might consider:

1. How do you derive the displacement function for a different simple harmonic motion?
2. What are the characteristics of simple harmonic motion?
3. Can you explain the physical significance of velocity being zero in this context?
4. How would you find the acceleration of the particle at \( t = \frac{\pi}{4} \)?
5. What changes if the amplitude or frequency of the motion is altered? 

**Tip:** When dealing with derivatives in physics, always remember to keep track of your units for clarity!

Find the velocity [w(t) = x'(t)] of the particle at time t = π /4 for the displacement function x(t) = 5 (sin (2t))^5.

Explore how to find the velocity of a particle described by the displacement function x(t) = 5(sin(2t))^5 at t = π/4. This step-by-step guide covers differentiation techniques, specifically the chain rule, suitable for students in Grades 11-12.

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