The location of a particle (in m) is given by its x, y, and z coordinates as function of the time (in s)?

Question

The location of a particle (in m) is given by its x, y and z coordinates as function of the time (in s) as:

x = 87-1t+1t2 and y = -5-23t-9t2 and z = -19-13t

a) Calculate the magnitude of the displacement from t = 6.00 s to t = 10.00 s.

b) Calculate the x-component of the instantaneous velocity at t = 3.00 s?

c) What is the magnitude of the object's acceleration at t = -1.00 s?

Answer 1

> a) Calculate the magnitude of the displacement from t = 6.00 s to t = 10.00 s.

Substitute "6.00" for t, in the three equations for x, y, z.  That will give you the displacement vector at t = 6 seconds.

Next, substitute "10.00" for t in the three equations for x, y, z.  That will give you the displacement vector at t = 10 seconds.

Finally, subtract the "t=6" vector from the "t=10" vector, using ordinary vector arithmetic (i.e., just subtract the coordinates individually). That will give you a third vector which is the displacement from t=6 to t=10.

> b) Calculate the x-component of the instantaneous velocity at t = 3.00 s?

First, differentiate the displacement with respect to time (i.e. differentiate each of the 3 functions x, y, z).  That will give you 3 coordinate formulas that tell you the particle's velocity vector as a function of time.

Next, just plug in "t=3" into the formula for the velocity vector's x-component.

> c) What is the magnitude of the object's acceleration at t = -1.00 s

Differentiate the velocity vector (the x, y, z formulas for velocity that you calculated in part (b)) one more time.  That will give you the vector function (3 coordinate equations) for acceleration.

Next, plug "t=-1" into the three equations to get the three components for acceleration at t=-1.  Now you have the acceleration vector for t=-1.

Finally, use the standard formula for finding the magnitude of a vector:

magnitude = sqrt(acc_x² + acc_y² + acc_z²)

Answer 2

The post modernistic answer would be that the it is pointless to answer this question.  A partical can never really be observed in it's actual state anyway, the simple act of observation changes it's perspective and effects the outcome of probability... :)

I don't think your professor would like that answer though but it could make them smile... if all else fails.