The position of an object moving along a line is given by p(t) = t - tsin(( pi )/4t) . What is the speed of the object at t = 7 ?

1 Answer
Surya K.
Feb 24, 2018

-2.18"m/s" is its velocity, and 2.18"m/s" is its speed.

Explanation:

We have the equation p(t)=t-tsin(pi/4t)

Since the derivative of position is velocity, or p'(t)=v(t), we must calculate:

d/dt(t-tsin(pi/4t))

According to the difference rule, we can write:

d/dtt-d/dt(tsin(pi/4t))

Since d/dtt=1, this means:

1-d/dt(tsin(pi/4t))

According to the product rule, (f*g)'=f'g+fg'.

Here, f=t and g=sin((pit)/4)

1-(d/dtt*sin((pit)/4)+t*d/dt(sin((pit)/4)))

1-(1*sin((pit)/4)+t*d/dt(sin((pit)/4)))

We must solve for d/dt(sin((pit)/4))

Use the chain rule:

d/dxsin(x)*d/dt((pit)/4), where x=(pit)/4.

=cos(x)*pi/4

=cos((pit)/4)pi/4

Now we have:

1-(sin((pit)/4)+cos((pit)/4)pi/4t)

1-(sin((pit)/4)+(pitcos((pit)/4))/4)

1-sin((pit)/4)-(pitcos((pit)/4))/4

That's v(t).

So v(t)=1-sin((pit)/4)-(pitcos((pit)/4))/4

Therefore, v(7)=1-sin((7pi)/4)-(7picos((7pi)/4))/4

v(7)=-2.18"m/s", or 2.18"m/s" in terms of speed.

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