The following statement is, in general, False for an object moving in more than
ID: 2877834 • Letter: T
Question
The following statement is, in general, False for an object moving in more than one dimension: "Acceleration is the derivative of the speed. " As shown by the steps below, the equation of motion vector r(t) = cos t cap i + sin t cap j is a counterexample to the statement in quotes above. Show that the speed of the motion given by vector r(t)is constant. Hence, any derivative of the speed is zero. Verify that for f the acceleration is non-zero for all t. For the given equation of motion, determine the geometrical relation between the acceleration and the velocity.Explanation / Answer
a)
r (t) = cos t i + sin t j
velocity,
v(t) = d/dt (r(t))
= -sin t i + cos t j
speed = |v(t)|
= sqrt (sin^2 t + cos ^2 t)
= sqrt (1)
= 1
It is consatant
b)
acceleration,
a = d/dt (v (t))
= d/dt (-sin t i + cos t j)
= -cos t i - sin t j
|a| = sqrt (sin^2 t + cos ^2 t)
= sqrt (1)
= 1
It is non zero
c)
a = dv/dt
acceleration is the derivative of velocity