Kinematics of a particle. (Chapter 12) презентация

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

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

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).

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

• Integrate acceleration for velocity and position.

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

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.

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

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

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.

0.0 m
6.0 m
18.0 m
9.0 m

50 m
100 m
150 m
200 m

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

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!).

v = 0.4 t

v = 12

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!).

48

30

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

8 s
4 s
10 s
6 s

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

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.

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!

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.

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.

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

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

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

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

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

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

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

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.

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

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.

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.

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

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.

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

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

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

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

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?

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

Plan:

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

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?

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.

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.

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

dsA/dt + dsB/dt = 0 ⇒ vB = -vA

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.

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!

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.

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.

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.

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).

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

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.

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).

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

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

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.

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!!

Example

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

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.