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Chapter 13 – Vector Functions13.4 Motion in Space: Velocity and Acceleration
13.4 Motion in Space: Velocity and Acceleration
Objectives: Determine how to
calculate velocity and acceleration.
Determine the motion of an object using the Tangent and Normal vectors.
13.4 Motion in Space: Velocity and Acceleration
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Position Vector Suppose a particle moves through space so that its
position vector at time t is r(t). Notice from the figure that, for small values
of h, the vector
approximates the direction of
the particle moving along the curve r(t).
Its magnitude measures the
size of the displacement vector
per unit time.
13.4 Motion in Space: Velocity and Acceleration
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Velocity VectorThe vector 1 gives the average velocity over
a time interval of length h and its limit is the velocity vector v(t) at time t :
The velocity vector is also the tangent vector and points in the direction of the tangent line.
13.4 Motion in Space: Velocity and Acceleration
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SpeedThe speed of the particle at time t is the
magnitude of the velocity vector, that is, |v(t)|.
For one dimensional motion, the acceleration of the particle is defined as the derivative of the velocity:
a(t) = v’(t) = r”(t)
| ( ) | | '( ) | = rate of change w.r.t. timeds
t tdt
v r
13.4 Motion in Space: Velocity and Acceleration
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VisualizationVelocity and Acceleration Vectors
13.4 Motion in Space: Velocity and Acceleration
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Example 1 Find the velocity, acceleration,
and speed of a particle with the given position function.
2( ) lnt t t t r i j k
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Newton’s Second Law of Motion
If the force that acts on a particle is known, then the acceleration can be found from Newton’s Second Law of Motion.
The vector version of this law states that if, any any time t, a force F(t) acts on an object of mass m producing an acceleration a(t), then
F(t) = ma(t)
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Example 2 – pg. 871 # 28A batter hits a baseball 3 ft
above the ground toward the center field fence, which is 10 ft high and 400 ft from home plate. The ball leaves the bat with speed 115 ft/s at an angle of 50o above the horizontal. Is it a home run? (Does the ball clear the fence?)
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Tangential and Normal Components of AccelerationWhen we study the motion of a
particle, it is often useful to resolve the acceleration into two components:
◦Tangential (in the direction of the tangent)
◦Normal (in the direction of the normal)
2'v v a T N
13.4 Motion in Space: Velocity and Acceleration
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Tangential and Normal Components of AccelerationWriting aT and aN for the tangential and
normal components of acceleration, we have a = aTT + aNN
whereaT = v’ and aN = v2
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Tangential and Normal Components of AccelerationWe will need to have aT = v’ and
aN = v2 in terms of r, r’, and r”. To obtain these formulas below, we start with v · a.
2
'( ) ''( )'
'( )
'( ) ''( )
'( )
T
N
t ta v
t
t ta v
t
r r
r
r r
r
13.4 Motion in Space: Velocity and Acceleration
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Example 3 – pg. 871 # 38Find the tangential and normal
components of the acceleration vector.
2( ) 1 2t t t t r i j
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Kepler’s LawsNote: Read pages 844 – 846.1. A planet revolves around the sun in an
elliptical orbit with the sun at one focus.
2. The line joining the sun to a planet sweeps out equal areas in equal times.
3. The square of the period of revolution of a planet is proportional to the cube of the length of the major axis of orbit.
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More Examples
The video examples below are from section 13.4 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 3◦Example 5◦Example 6
13.4 Motion in Space: Velocity and Acceleration
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Demonstrations
Feel free to explore these demonstrations below.
Kinematics of a Moving PointBallistic Trajectories
