A particle is confined to move in one dimension. For times t > 0 the particle\'s

Question

A particle is confined to move in one dimension. For times t > 0 the particle's position is given by r = R(1 - e^-kt) r where r is measured in meters and t is measured in seconds. R and k are constants. (a) What are the dimensions of the constants in Eq. 4? (b) Determine the velocity and acceleration of the particle for all times t > 0. (c) What are the limiting values of r, v, and a as t rightarrow 0 and t rightarrow infinity? (d) If R = 10 and kappa = 0.25 in appropriate units, where is the particle when the acceleration has half its maximum value? What is the velocity at this location? A point mass is moving according to the position function r = A cos (w t) i + 2 A sin (w t) j (a) Construct a parametric plot of the path followed by the mass through the given i, j space. Describe this path. (b) Determine the velocity and acceleration of the point mass. Construct parametric plots of velocity and acceleration. What are the dimensions of the axes on these two plots? (c) Find the points on the path where the speed is not changing. Two objects follow circular paths about a common center. In polar coordinates the first object has a velocity of upsilon_1 = 50 m/s Phi at a radius of 20 m. The second object has a velocity of upsilon_2 = 38 m/s Phi at a radius of 35 m. Both velocities are measured relative to the common center point. (a) Determine the angular speed and centripetal accelerations of each object. (b) Write expressions for the position, velocity, and acceleration for each object relative to the other. Assume that for both objects Phi_0 = 0 when t = 0. (c) Construct plots of the relative upsilon_rho and upsilon_ found in (b) for each object. Your plots should cover at least two full orbits of object 2. Does either object manifest "retrograde motion" apparent motion in the - direction)? Comment on the significance of retrograde motion to the work of Tycho Brahe and Johannes Kepler.

Explanation / Answer

Given,

r = R (1 - e^-kt)

a)The dimension of R should be m and that of k should be s-1(per second)

b)we know that,

v = dr/dt = d[R (1 - e^-kt)]/dt

v = k R e^-kt

we know that acceleration is:

a = dv/dt = d (k R e^-kt)/dt

a = k^2 R (-e^-kt)

c)when t -->0

e^0 = 1

r = R ( 1 - 0) = 0 ; v = k R (1) = k R ; a = k^2 R (-1)

r = 0 ; v = k R ; a = -k^2 R

and t--> infinity

e^infinity = infinity

all of the three quanitities will be infinite.

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