Kinematics of a particle. (Chapter 12)

and acceleration) of a particle traveling along a straight path.

can often be analyzed as if they were particles.
Why?

If we measure the altitude of this rocket as a function of time, how can we determine its velocity and acceleration?

the car as a particle?

If the car accelerates at a constant rate, how can we determine its position and velocity at some instant?

coordinate axis s.

The total distance traveled by the particle, sT, is a positive scalar that represents the total length of the path over which the particle travels.

The position of the particle at any instant, relative to the origin, O, is defined by the position vector r, or the scalar s. Scalar s can be positive or negative. The typical unit is meter (m).

position of a particle. It is a vector quantity (it has both magnitude and direction). The magnitude of the velocity is called speed, with unit m/s.

particle. It is a vector quantity. Typical units are m/s2.

As the text indicates, the derivative equations for velocity and acceleration can be manipulated to get: a ds = v dv

= ds/dt ; a = dv/dt or a = v dv/ds

• Integrate acceleration for velocity and position.

• Note that so and vo represent the initial position and velocity of the particle at t = 0.

case when acceleration is constant (a = ac) to obtain very useful equations. A common example of constant acceleration is gravity; i.e., a body freely falling toward earth. In this case, ac = g = 9.81 m/s2 downward. These equations are:

traveling. Since the velocity is given as a function of time, take a derivative of it to calculate the acceleration. Conversely, integrate the velocity function to calculate the position.

Given: A particle travels along a straight line to the right with a velocity of v = ( 4 t – 3 t2 ) m/s where t is in seconds. Also, s = 0 when t = 0.

Find: The position and acceleration of the particle
when t = 4 s.

= dv / dt = d(4 t – 3 t2) / dt = 4 – 6 t
⇒ a = – 20 m/s2 (or in the ← direction) when t = 4 s

or "CH" button. 2. While the light is flashing red and green, enter the 2 digit channel code (i.e. channel 1 = 01, channel 21 = 21). Channel is 11 3. After the second digit is entered, Press and release the "GO" or "CH" button. The light should flash green to confirm. 4. Press and release the "1/A" button. The light should flash amber to confirm.

with time as shown. The average acceleration of the particle is?

0.4 m/s2
0.4 m/s2
1.6 m/s2
1.6 m/s2

left. If it then passes through the same location 5 seconds later with a velocity of 50 m/s to the right, the average velocity of the particle during the 5 s time interval is?

10 m/s
40 m/s
16 m/s
0 m/s

its velocity is defined as v = (-4s2) m/s, where s is in meters.

Find: The velocity and acceleration as functions of time if s = 2 m when t = 0.

Plan: Since the velocity is given as a function of distance, use the equation v=ds/dt.
1) Express the distance in terms of time.
2) Take a derivative of it to calculate the velocity and acceleration.

acceleration.

left at s0 = 0 m. Determine its position when t = 3 s if the acceleration is 2 m/s2 to the right.

0.0 m
6.0 m
18.0 m
9.0 m

12 m/s and constant acceleration of 3.78 m/s2 in the same direction as the velocity. Determine the distance the particle has traveled when the velocity reaches 30 m/s.

50 m
100 m
150 m
200 m

beautiful aircraft
3 = very beautiful
4 = pretty beautiful
5 = beautiful
6 = ugly
7 = pretty ugly
8 = very ugly
9 = extremely ugly aircraft
10 = most ugly aircraft ever built

particle using graphs.

are linked and that integration can be thought of as finding the area under a curve.

under the v-t graph during time Δt.

v-s graph. By reading the velocity v at a point on the curve and multiplying it by the slope of the curve (dv/ds) at this same point, we can obtain the acceleration at that point. Recall the formula
a = v (dv/ds).
Thus, we can obtain an a-s plot from the v-s curve.

the s-t graph at key points. What are those?
when 0 < t < 5 s; v0-5 = ds/dt = d(3t2)/dt = 6 t m/s
when 5 < t < 10 s; v5-10 = ds/dt = d(30t−75)/dt = 30 m/s
v-t graph

at various points along the v-t graph. Using the results of the first part where the velocity was found:
when 0 < t < 5 s; a0-5 = dv/dt = d(6t)/dt = 6 m/s2
when 5 < t < 10 s; a5-10 = dv/dt = d(30)/dt = 0 m/s2
a-t graph

graph shown, the particle’s velocity at t = 20 s is

200 m/s
100 m/s
0
20 m/s

s
20 s
30 s
40 s

traveled for the 0 - 50 s interval.
Plan: What is your plan?

traveled for the 0 - 50 s interval.

Plan: Find slopes of the v-t curve and draw the a-t graph.
Find the area under the curve. It is the distance traveled.
Finally, calculate average speed (using basic definitions!).

dv/dt = 0.4 m/s²
For 30 ≤ t ≤ 50 a = dv/dt = 0 m/s²
a-t graph

0.4 t dt = 0.4 (1/2) (30)2 = 180 m
Δs30-50 = ∫ v dt
= ∫ 12 dt = 12 (50 – 30)
= 240 m
s0-50 = 180 + 240 = 420 m
vavg(0-50) = total distance / time
= 420 / 50
= 8.4 m/s

v = 0.4 t

v = 12

traveled for the 0 - 48 s interval.
Plan:

Plan: Find slopes of the v-t curve and draw the a-t graph.
Find the area under the curve. It is the distance traveled. Finally, calculate average speed (using basic definitions!).

dv/dt = 0.2 m/s²
For 30 ≤ t ≤ 48 a = dv/dt = -0.333 m/s²
a-t graph

(30)2 = 90 m
Δs30-48 = ∫ v dt = [(-1/3) (1/2) (t – 48)2]
= (-1/3) (1/2)(48 – 48)2 – (-1/3) (1/2)(30 – 48)2
= 54 m
s0-48 = 90 + 54 = 144 m
vavg(0-48) = total distance / time
= 144 m/ 48 s
= 3 m/s

48

30

of 2 m/s. The acceleration is given as a = 30 v-4 [m/s2]
Find: Determine the velocity and the distance covered after 40 s
Plan:

Plan: For the determining of the velocity acknowledge the fact that the acceleration has been given as a function of velocity
=> use a = dv/dt
For determining the distance acknowledge that we are now looking for distance s while a has been given
=> use a ds = v dv

t necessary for the car to travel 100 meters.

8 s
4 s
10 s
6 s

along a curved path.

radar and its x, y, and z coordinates (relative to a point on earth) recorded as a function of time.

How can we determine the velocity or acceleration of the plane at any instant?

a constant speed.

motion. Since the motion is often three-dimensional, vectors are used to describe the motion.

The position of the particle at any instant is designated by the vector
r = r(t). Both the magnitude and direction of r may vary with time.

particle.

The average velocity of the particle during the time increment Δt is
vavg = Δr/Δt .
The instantaneous velocity is the time-derivative of position
v = dr/dt .
The velocity vector, v, is always tangent to the path of motion.

The magnitude of v is called the speed. Since the arc length Δs approaches the magnitude of Δr as t→0, the speed can be obtained by differentiating the path function (v = ds/dt). Note that this is not a vector!

particle.

If a particle’s velocity changes from v to v’ over a time increment Δt, the average acceleration during that increment is:
aavg = Δv/Δt = (v’ -v )/Δt
The instantaneous acceleration is the time-derivative of velocity:
a = dv/dt = d2r/dt2

A plot of the locus of points defined by the arrowhead of the velocity vector is called a hodograph.
The acceleration vector is tangent to the hodograph, but not, in general, tangent to the path function.

rectangular components of the vectors.

particle in terms of its x, y, z or rectangular components, relative to a fixed frame of reference.

 

the path of the particle.

 

 

= (0.05 x2) m, where x is in meters.
vx = -3 m/s, ax = -1.5 m/s2 at x = 5 m.

Find: The y components of the velocity and the acceleration of the box at x = 5 m.

Plan: Note that the particle’s velocity can be related by taking the first time derivative of the path’s equation. And the acceleration can be related by taking the second time derivative of the path’s equation.
Take a derivative of the position to find the component of the velocity and the acceleration.

+ 1) i + (4t – 1) j ] (m), its speed at t = 1 s is

2 m/s
3 m/s
5 m/s
7 m/s

i - 2x j) m. Determine the particle’s acceleration.

(4 i +8 j ) m/s2
(8 i -16 j ) m/s2
(8 i) m/s2
(8 j ) m/s2

she can project water from the hose. What parameters would you program into a wrist computer to find the angle, θ, that she should use to hold the hose?

motions, one in the horizontal direction experiencing zero acceleration and the other in the vertical direction experiencing constant acceleration (i.e. from gravity).

horizontal direction remains constant (vx = vox) and the position in the x direction can be determined by:
x = xo + (vox) t

Why is ax equal to zero (what assumption must be made if the movement is through the air)?

= – g. Application of the constant acceleration equations yields:
vy = voy – g t
y = yo + (voy) t – ½ g t2
vy2 = voy2 – 2 g (y – yo)

For any given problem, only two of these three equations can be used. Why?

relations in x- and y-directions.

Since y = 0 at C
0 = (10 sin 30) t – ½ (9.81) t2 ⇒ t = 0 and 1.019 s

t
x = (10 cos 30) 1.019 = 8.83 m

of the coordinate system is placed at A). Apply the kinematic relations in x- and y-directions.

with initial velocity Vo at angle θ, is equal to?

(vo sin θ)/g
(2vo sin θ)/g
(vo cos θ)/g
(2vo cos θ)/g

Components

velocity and acceleration of a particle traveling along a curved path.

can be approximated by a function
y = f(x).
The roller coaster starts from rest and increases its speed at a constant rate.

How can we determine its velocity and acceleration at the bottom?

it is sometimes convenient to describe its motion using coordinates other than Cartesian. When the path of motion is known, normal (n) and tangential (t) coordinates are often used.

In the n-t coordinate system, the origin is located on the particle (the origin moves with the particle).

The t-axis is tangent to the path (curve) at the instant considered, positive in the direction of the particle’s motion.
The n-axis is perpendicular to the t-axis with the positive direction toward the center of curvature of the curve.

instant is defined by the distance, s, along the curve from a fixed reference point.

the path of motion (t-direction).

acceleration vector:
a = at ut + an un

The normal or centripetal component is always directed toward the center of curvature of the curve. an = v2/r

 

 

consider.

at = (at)c.
In this case, s = so + vo t + (1/2) (at)c t2
v = vo + (at)c t
v2 = (vo)2 + 2 (at)c (s – so)
As before, so and vo are the initial position and velocity of the particle at t = 0. How are these equations related to projectile motion equations? Why?

and t axes are defined as before. At any point, the t-axis is tangent to the path and the n-axis points toward the center of curvature. The plane containing the n and t axes is called the osculating plane.

A third axis can be defined, called the binomial axis, b. The binomial unit vector, ub, is directed perpendicular to the osculating plane, and its sense is defined by the cross product ub = ut × un.

In our cases there is no motion, thus no velocity or acceleration, in the binomial direction.

= (2s) m/s, where s is in meters. r = 50 m

Find: The magnitudes of the car’s acceleration at s = 10 m.
Plan:

1) Calculate the velocity when s = 10 m using v(s).
2) Calculate the tangential and normal components of acceleration and then the magnitude of the acceleration vector.

magnitude is given by v = (2s) m/s. When s = 10 m: v = 20 m/s

 

 

Normal component: an = v2/r
When s = 10 m: an = (20)2 / (50) = 8 m/s2

The magnitude of the acceleration is
a = [(at)2 + (an)2]0.5 = [(40)2 + (8)2]0.5 = 40.8 m/s2

at a speed that increases with time,
v = (0.0625 t2) m/s.

Find: The magnitudes of the boat’s velocity and acceleration at the instant t = 10 s.
Plan:

The boat starts from rest (v = 0 when t = 0).
1) Calculate the velocity at t = 10 s using v(t).
2) Calculate the tangential and normal components of acceleration and then the magnitude of the acceleration vector.

magnitude is given by
v = (0.0625t2) m/s. At t = 10s:
v = 0.0625 t2 = 0.0625 (10)2 = 6.25 m/s

 

 

Normal component: an = v2/r m/s2
At t = 10 s: an = (6.25)2 / (40) = 0.9766 m/s2

The magnitude of the acceleration is
a = [(at)2 + (an)2]0.5 = [(1.25)2 + (0.9766)2]0.5 = 1.59 m/s2

has an instantaneous velocity of 30 m/s and its velocity is increasing at a constant rate of 4 m/s2. What is the magnitude of its total acceleration at this instant?

3 m/s2
4 m/s2
5 m/s2
-5 m/s2

and an acceleration of 14 m/s2 acting in the direction shown.

Find: The rate of increase in the train’s speed and the radius of curvature ρ of the path.
Plan:

Determine the tangential and normal components of the acceleration
Calculate dv/dt form the tangential component of the acceleration
Calculate ρ from the normal component of the acceleration

m/s2
Normal component :
an = 14 sin(75) = 13.52 m/s2

The normal component of acceleration is
an = v2/r ⇒ 13.52 = 202 / r
r = 29.6 m

The tangential component of the acceleration is the rate of increase of the train’s speed so
at = 3.62 m/s2

coordinates.

moves along a 3-D curve.

In the figure shown, the box slides down the helical ramp. How would you find the box’s velocity components to check to see if the package will fly off the ramp?

as r = r ur. Note that the radial direction, r, extends outward from the fixed origin, O, and the transverse coordinate, θ, is measured counter-clockwise (CCW) from the horizontal.

position can be written as
rP = rur + zuz
Taking time derivatives and using the chain rule:

any instant, its angular position is θ = (4t3/2) rad, where t is in seconds.
A ball rolls outward so that its position is r = (0.1t3) m.

Find: The magnitude of velocity and acceleration of the ball when t = 1.5 s.
Plan:

Two Particles

particles undergoing dependent motion.

heavy objects. The total lifting force required from the truck depends on both the weight and the acceleration of the cabinet.

How can we determine the acceleration and velocity of the cabinet if the acceleration of the truck is known?

the speed of the mine car, A, relative to the speed of the motor, M.
It is important to establish the relationships between the various motions in order to determine the power requirements for the motor and the tension in the cable.

For instance, if the speed of the cable (P) is known because we know the motor characteristics, how can we determine the speed of the mine car? Will the slope of the track have any impact on the answer?

depend on the motion of another object.

The motion of each block can be related mathematically by defining position coordinates, sA and sB. Each coordinate axis is defined from a fixed point or datum line, measured positive along each plane in the direction of motion of each block.

defined from fixed datum lines extending from the center of the pulley along each incline to blocks A and B.

If the cord has a fixed length, the position coordinates sA and sB are related mathematically by the equation
sA + lCD + sB = lT
Here lT is the total cord length and lCD is the length of cord passing over the arc CD on the pulley.

incline (positive sA direction), B moves up the incline (negative sB direction).

Accelerations can be found by differentiating the velocity expression. Prove to yourself that aB = -aA .

dsA/dt + dsB/dt = 0 ⇒ vB = -vA

defined from fixed datum lines, measured along the direction of motion of each block.

Note that sB is only defined to the center of the pulley above block B, since this block moves with the pulley. Also, h is a constant.

The red colored segments of the cord remain constant in length during motion of the blocks.

sA = lT
Where lT is the total cord length minus the lengths of the red segments.

Since lT and h remain constant during the motion, the velocities and accelerations can be related by two successive time derivatives:
2vB = -vA and 2aB = -aA

When block B moves downward (+sB), block A moves to the left (-sA). Remember to be consistent with your sign convention!

for B (sB) from the bottom pulley instead of the top pulley.

The position, velocity, and acceleration relations then become
2(h – sB) + h + sA = lT
and 2vB = vA 2aB = aA

Prove to yourself that the results are the same, even if the sign conventions are different than the previous formulation.

motion of particles moving along rectilinear paths (only the magnitudes of velocity and acceleration change, not their line of direction).

4. Differentiate the position coordinate equation(s) to relate velocities and accelerations. Keep track of signs!

3. If a system contains more than one cord, relate the position of a point on one cord to a point on another cord. Separate equations are written for each cord.

2. Relate the position coordinates to the cord length. Segments of cord that do not change in length during the motion may be left out.

1. Define position coordinates from fixed datum lines, along the path of each particle. Different datum lines can be used for each particle.

is pulled down with a speed of 2 m/s.

Find: The speed of block B.
Plan:

There are two cords involved in the motion in this example. There will be two position equations (one for each cord). Write these two equations, combine them, and then differentiate them.

fixed position).
sA can be defined to the point A.
sB can be defined to the center of the pulley above B.
sC is defined to the center of pulley C.
All coordinates are defined as positive down and along the direction of motion of each point/object.

1) Define the position coordinates from a fixed datum line. Three coordinates must be defined: one for point A (sA), one for block B (sB), and one for block C (sC).

= l1 + 2l2

2) Write position/length equations for each cord. Define l1 as the length of the first cord, minus any segments of constant length. Define l2 in a similar manner for the second cord:

4) Relate velocities by differentiating this expression. Note that l1 and l2 are constant lengths.
vA + 4vB = 0 ⇒ vB = – 0.25vA = – 0.25(2) = – 0.5 m/s
The velocity of block B is 0.5 m/s up (negative sB direction).

Cord 1: sA + 2sC = l1
Cord 2: sB + (sB – sC) = l2

the above.

speed of 4 m/s while block C is moving up at 2 m/s.

Find: The speed of block B.
Plan:

All blocks are connected to a single cable, so only one position/length equation will be required. Define position coordinates for each block, write out the position relation, and then differentiate it to relate the velocities.

can be written:
sA + 2sB + sC = l

3) Differentiate to relate velocities:
vA + 2vB + vC = 0
4 + 2vB + (-2) =0
vB = -1 m/s

1) A datum line can be drawn through the upper, fixed, pulleys and position coordinates defined from this line to each block (or the pulley above the block).

The velocity of block B is 1 m/s up (negative sB direction).

down at 6 m/s while block C is moving down at 18 m/s.

24 m/s
3 m/s
12 m/s
9 m/s

Particles
Using Translating Axes

particles undergoing relative motion.

got lost.
The fighter pilot needs to make sure he does intercept the airliner at the correct location at the correct altitude

own path. Their absolute position vectors are rA and rB when measured from the position of the fixed observer.

There is a second reference frame x’-y’-z’ that is moving with respect to the observer at O but is fixed to particle A. This reference frame is only allowed to translate with respect to the fixed reference frame.

The position of B can be measured relative to A with the relative-position vector rB/A. The following relation holds:
rB = rA + rB/A

vB = vA + vB/A
And for the acceleration:
aB = aA + aB/A

numerically using trigonometry or by resolving each of the three vectors into a coordinate system and thereby generating a set of scalars.

2. Since vector addition forms triangles there can be at most two unknowns. The represent magnitudes and/or directions of the vector quantities.

1. First specify the particle A that is the origin for the translating x’, y’, z’-axes. Usually this point has a known velocity and/or acceleration.

with different speeds and in different directions.

Find: What is the distance between them after 4 seconds and what is the direction of boat B with respect to boat A?
Plan:

The origin of the x- and y-axes are located at O. First determine the positions of A and B after 4 seconds. Then use relative positions to find the position of B with respect to A. Use vectors!!

= 10 cos 60° i + 10 sin 60° j
After 4 seconds
rA = 60 cos 30° i + 60 sin 30° j
rB = 40 cos 60° i + 40 sin 60° j
Then use rB = rA + rB/A

Example

 

the magnitude of the velocity of plane B relative to plane A

693 km/h
650 km/h
400 km/h
1258 km/h

fighter B is flying along a circular path with a radius of curvature of 400 km.

Find: Determine the velocity and acceleration of fighter B as measured by the pilot of aircraft A.
Plan:

The origin of the x- and y-axes are located in an arbitrary but fixed point. The translating reference frame is attached to A. Then apply the relative velocity and relative acceleration in scalar form. This is done because at the instant of measuring both reference frames are parallel.

+ vB/A
600 j km/h = 700 j km/h + vB/A
vB/A = -100 j km/h angle 270°
2) For the acceleration: Fighter B has both normal and tangential accelerations since it is flying a curved path.
(aB)t = -100 j km/h2
(aB)n = vB2 /ρ = (600 km/h)2/400 km = 900i km/h2
Thereby:
aB = aA + aB/A => 900i – 100j = 50j + aB/A
Thus:
aB/A = (900i -150j) km/h2 angle = 350.5°