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1. Match each of the vector valued functions to its graph.
(A) r(t) = ⟨ 8 cos(4t), 8 sin(4t),t⟩ (B) r(t) = ⟨ 8 cos(4t),t, 8 sin(4t)⟩
(C) r(t) = ⟨ t cos(4t),t,tsin(4t)⟩ (D) r(t) = ⟨ 2 + t, 0, 2t⟩
(E) r(t) = ti + tj + (2 − t
2
)k (F) r(t) = sint
2
i + 5 cost
2
j + tk
(I) (II)
(III) (IV )
(V ) (V I)
2
2
2
2
Ï€
2. Find the curvature κ of the curve →r(t) = ⟨ 2 sin t, 3t, 2 cost⟩ when t =
Ï€
.
3. The parametric equations of a moving point are
x(t) = 2 sin 3t, y(t) = −8t, z(t) = 2 cos 3t.
(a) Find its velocity, speed, and acceleration at all times.
(b) Find the distance traveled by the point from t = 0 to t = 5.
(c) Find the equation of the tangent line to the curve at the point when t = 0.
4. A body moves in space so that at time t the acceleration of the body is given by the
vector a(t) = (cos t)i + (sin t)j − gk, where g is a constant scalar value. At t = 0 the
body is at the origin and has velocity 4j.
(a) What is the position of the body when t =
Ï€
?
(b) What is the position of the body when t = π?
5. Batman drives his rocket powered car over the edge of a cliff. His initial position
at the edge of the cliff is P = (0, 80). His initial velocity vector is v(0) = 16i. His
acceleration vector is given by a(t) = 20i − 10j.
(a) Find Batman’s position vector r(t).
(b) How long before Batman crashes into the x-axis (i.e. the ground)?
(c) What is Batman’s speed when he hits the ground?
Projectile Motion, More Curvature, Tangential and Normal Components of Acceleration
6. A ball is thrown at an angle of 45o
to the ground. If the ball lands 90 m away, what
was the initial speed of the ball?
7. A canon is fired with an angle of elevation of 45 degrees. The speed of the canonball
as it leaves the muzzle is 128 ft/sec. On Earth the acceleration due to gravity is
32 ft/sec2 downward.
(a) What is the maximum height of the cannonball?
(b) How long is the canonball be in the air?
8. Find the curvature κ, and the tangential and normal components of acceleration, aT
and aN , of the curve →r(t) = ⟨ 2 sin t, 3t, 2 cost⟩ when t = .
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