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 #?

Question

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 #?

Nora Arzate · Accepted Answer

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

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.

Elena Albarran · Answer

undefined Calculate the derivative of the position function with respect to time and evaluate it at t = 7 to find the speed of the object. The speed is the magnitude of the derivative.