Expert: Ahmed Salami - 11/12/2010

Question
Consider a car shock absorber modeled by the equation h(t)=e^-0.5t sint,
where h(t) reperesents the vertical displacement in metres, t represents
time in seconds.

A) determine when the maximum vertical velocity , in metres per second,
occurs and its value
b) the greatest force felt by passengers is felt when acceleration is at a
maximum.  Use F=ma to determine the greatest force felt by a passenger
who is 60kg, where F is force in newtons, m is mass in kilograms, and a is
acceleration in m/s^2.

A) I took the first derivative of h(t) to get velocity equation and then took
the derivative of that and set it to zero to find the time the maximum
velocity occurs and integrated it into the velocity equation but didn't get the
correct answer of time being zero ...

B) for this one, I tookthe derivative of the velocity equation because it is
equal to acceleration equation and then took the derivative of that and set
it to zero to find the max acceleration then substituted the value into F=ma
but I didn't get the right answer agaain... The answer should be 12.7
newtons but I can't get it.

Could you show me how you would do these questions so I can compare
my answers?

Hope this makes sense and you can help!  

Answer
Hi Erin,
Well, from a practical point of view, the vertical velocity should be maximum at t = 0 and should decrease to zero, that is the idea of a shock absorber.

a) Now, did you arrive at tan t = -4/3 with one maximum occurring at t = -0.9273 ?
You need to realize that the practical domain of t is only the set of positive real numbers and it's at t = 0 that we have the maximum velocity (of 1 m/s). I have included a graphical image that should show you more.

b) For this one, did you arrive at tan t = 2/11 with one maximum occurring at t = -2.962 ?
For t > 0, a maximum acceleration occurs at t = 3.32 with a value of about 0.212 m/s². Again, the graph explains it.
Therefore, F = 60 * 0.212 = 12.72 N

Keep in mind that the equations for displacement, velocity and acceleration are oscillatory and have more than one optimum points. Its up to you then to select appropriate ones as related to your requirements.

You can always get back to me.

Regards and best of luck.

Edit: Added link to graph.
http://www.speedyshare.com/files/25180253/Shock.png