A moving particle starts at an initial position r(0)= (1,0,0) with initial velocity (0)=(-/+k. its acceleration is a(t) = 4tl+ 6tj + k. Determine its velocity and position when t = 2
Answers
Answer:
iven by
v(t) = h 2t, et
, 3 cost i
(1) Find the position vector r(t) of the particle.
Recall: Let the position of an object be given by r = hf(t), g(t), h(t)i, for t ≥ 0 . The velocity of the
object is v(t) = hf
0
(t), g0
(t), h0
(t)i. If v(t) = hx(t), y(t), z(t)i
r(t) = Z
x(t) dt + C1,
Z
y(t) dt + C2,
Z
z(t) dt + C3
r(t) = Z
2t dt + C1,
Z
e
t
dt + C2,
Z
3 cost dt + C3
=
t
2 + C1, et + C2, 3 sin t + C3
Since r(0) = *
0
2 + C1
| {z }
C1
, e0 + C2
| {z }
1+C2
, 3 sin 0 + C3
| {z }
C3
+
= h0, 1, 0i, we have C1 = C2 = C3 = 0.
The position vector r(t) of the particle is
r(t) =
t
2
, et
, 3 sin t
(2) Find the acceleration a(t) of the particle.
Recall: If the velocity of the object is v(t) = hx(t), y(t), z(t)i, then the acceleration of the object is
a(t) = hx
0
(t), y0
(t), z0
(t)i
The acceleration a(t) of the particle is
v(t) = h (2t)
0
, (e
t
)
0
, (3 cost)
0
i = h 2, et
, −3 sin t i
(3) Which of the following points are on the curve of r
( 1, 2, 3 ), ( 4, e2
, 3 sin(2) ), ( 0, 0, 0 )?
If some point is on the curve, find the speed of the particle at that point.
Recall: A point (a, b, c) in on the curve of r = hf(t), g(t), h(t)i if there exists t such that
(a, b, c) = (f(t), g(t), h(t)).
A point (a, b, c) in on the curve of r(t) = ht
2
, et
, 3 sin ti if there exists t such that
(a, b, c) = (t
2
, et
, 3 sin t)
• If (1, 2, 3) = (t
2
, et
, 3 sin t), then 1 = t
2
, 2 = e
t
, and 3 = 3 sin t. The equation
1 = t
2
implies t = 1, but 2 6= e
1 = e. The point (1, 2, 3) is not on the curve of r(t).
• If ( 4, e2
, 3 sin(2) ) = (t
2
, et
, 3 sin t), then 4 = t
2
, e
2 = e
t
, and 3 sin(2) = 3 sin t.
For t = 2 all three equations are satisfied. The point ( 4, e2
, 3 sin(2) ) is on the
curve of r(t).
• If (0, 0, 0) = (t
2
, et
, 3 sin t), then 0 = t
2
, and 0 = e
t
, and 0 = 3 sin t. The equation
0 = t
2
implies t = 0, but 0 6= e
0 = 1. The point (0, 0, 0) is not on the curve of r(t).
the speed of the object
to the curve of r(t)
at time t = 2, i.e.
at the point ( 4, e2
, 3 sin(2) )
= |v(2)| = |h 2 · 2, e2
, 3 cos 2 i|
=
p
4
2 + (e
2
)
2 + (3 cos 2)2 =
√
16 + e
4 + 9 cos2 2
Step-by-step explanation: