vfc-a-20
Post on 28-Sep-2015
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Vector Functions and Curves
One variable functions
Question
An object travels on the curve of intersection of the cylinders y = x2 andz = x2 with increasing x. When the particle is at (1,1, 1), it has a speed of9cm/s which is increasing at a rate of 3cm/s2. Given that all distances arein cm, find the velocity and acceleration of the object at that point.Answer
r = xi x2j + x2kv =
dx
dt(i 2xj + 2xk)
a =d2x
dt2(i 2xj + 2xk) +
(dx
dt
)2(2j + 2k).
|v| =
dxdt1 + 4x4 + 4x4
=1 + 8x4
dx
dt,
as x is increasing.At (1,1, 1), x = 1 and |v| = 9, dx
dt= 3.
at that pointv = 3i 6j + 6k
.Now
d
dt|v| =
1 + 8x4
d2
dt2+
16x31 + 8x4
(dx
dt
)2.
When x = 1, the left side is 3.
3
(d2x
dt2
)+ 48 = 3
andd2x
dt2= 15
at that point. Acceleration at that point
a = 15(i 2j + 2k) + 9(2j + 2k)= 15i + 12j 12k
1
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