Hello Everyone, please help me on this dynamics questions!
A particle has an initial velocity U and after travelling a distance d in time T along a straight path its velocity is V. The retardation of the particle is proportional to its velocty at any time. Show that:
U = V e ^ ( T(U-V)/d )
Hello ModernizedIn this problem we use two different ways to express acceleration; first in terms of time; secondly in terms of distance. They are:
Acceleration = .
The first part of this problem connects velocity, time, and acceleration, so we express the phrase:in the formThe retardation of the particle is proportional to its velocty at any time
where is a positive constant
, noting that the boundary conditions have been incorporated into the limits of the integrals.
... (1)
Next we use the alternative expression for acceleration, in order to relate velocity, acceleration and distance:
, where is a positive constant.
, noting again the boundary conditions as limits.
From here we can find another expression for , in terms of V, U and d (it's very easy!). And hence, from equation (1), we can get the result.
Can you complete it now?
Grandad