Consider s(t)=5sin(t) describes a block bouncing on a spring. Complete the table with average velocities. Make the conjecture about the value of the instantaneous velocity at t=pi/2?

[ #pi/2#, #pi#] [ #pi/2#, #pi/2#+0.1][ #pi/2#, #pi/2#+0.01]

1 Answer
Jun 13, 2017

Answer:

  1. #v_"av" = -3.18# #"m/s"#

  2. #v_"av" = -0.250# #"m/s"#

  3. #v_"av" = -0.0250# #"m/s"#

#v(pi/2) = 0# #"m/s"#

Explanation:

We're asked to calculate the average velocity for three different time intervals, and the instantaneous velocity at #t = pi/2# #"s"#.

1.

For the #sfcolor(red)("first"# average velocity, we'll plug in the two times #t = pi/2# #"s"# and #t = pi# #"s"# into the position equation to find the position at those times:

#s(pi/2) = 5sin(pi/2) = 5# #"m"#

#s(pi) = 5sin(pi) = 0# #"m"#

Next, we'll use the average velocity formula

#v_"av" = (Deltax)/(Deltat)#

The quantity #Deltax# is the change in position of the object, which is

#0"m" - 5"m" = -5"m"#.

(I'm assuming the distance units are meters.)

The quantity #Deltat# is the time interval, which is

#pi# #"s"# # - pi/2# #"s" = pi/2# #"s"#

The average velocity is thus

#v_"av" = (-5"m")/(pi/2"s") = -3.18# #"m/s"#

2.

Similarly, we'll plug in the times and find the position, then use this to find the average velocity:

#s(pi/2) = 5sin(pi/2) = 5# #"m"#

#s(pi/2 + 0.1) = 5sin(pi/2 + 0.1) = 4.975# #"m"#

#Deltax = 0.025# #"m"#

#Deltat = 0.1# #"s"#

#v_"av" = (Deltax)/(Deltat) = -0.250# #"m/s"#

3.

#s(pi/2) = 5sin(pi/2) = 5# #"m"#

#s(pi/2 + 0.01) = 5sin(pi/2 + 0.01) = 4.99975# #"m"#

#Deltax = -0.000249998# #"m"#

#Deltat = 0.01# #"s"#

#v_"av" = -0.0250# #"m/s"#

4.

As we can see, if we keep plugging in smaller time intervals, the average velocity gets closer and closer to #0#. We can then conclude that the instantaneous velocity of the block at #t = pi/2# #"s"# is #color(red)(0# #color(red)("m/s"#