3 Ch 03 Kinematics in two Dimensions.pdf

 (1) Motion in the initial direction with constant velocity and,

(1) The magnitude of the velocity vector (speed) may change

(2) Only the direction of the velocity vector may change

(2) The motion of a particle freely falling in the vertical

(3) Both the magnitude and direction of the velocity vector

(a) A woman walking across a railroad car.

(a) Both stones strike the water with the same velocity. (b) Stone 1 strikes with the

(a) x and v x,

(a)From the kinematic equations,

(a)the initial speed of the

(b) Vector diagram for the velocity of the woman relative to the

(b) y and v y, and

(b)the height of the cliff.

(b)What should his speed have been at the

(c) the final velocity of the spacecraft.

(c) the final velocity of the spacecraft.

(c)If his speed was only half the value

(vo sinθo)t =(vo sinθo) xT / (vocosθo) = xT tanθo

∴ time taken to reach the peak is: t1 = (vo sinθo)/g

● Also, the y component of velocity is zero at the peak of the path.

2. Decide which directions are to be called positive (+) and

A 45 o launch angle gives the greatest

A clever monkey escapes from the zoo. The zookeeper finds him in a tree. Failing to

A daring swimmer dives

A grasshopper leaps into

A particle moving in the x-y plane is located with the position vector r drawn

A physics professor did daredevil stunts in

A placekicker kicks a football at and angle of 40.0 degrees and the initial

A player kicks a football at an angle of 40.0° from the

A power boat, starting from rest, maintains a constant acceleration. After a certain

A projectile fired from the origin at t = 0 with an initial

A projectile fired from the origin with an initial speed vo at

a y = g = 9.80 m/s 2.

a. 2r and 2v

a. behind you,

above the horizontal.

Absence of air resistance

Acceleration must now be

across a river that is 1800 m

always hits the monkey, regardless of the dart’s muzzle velocity (provided that it

and 9.00 m below the top

and the dart have the same x- and y-coordinates. First, let’s ask when the x-coordinates are the same. The

and the horizontal range is R. At the peak of the trajectory

choose our reference frame such that the y-direction

component of +22 m/s and an acceleration of +24

Conceptual Example:

Conceptual Example: I Shot a Bullet into the Air...

considered during change

considered during change

constitute acceleration of a particle.

Crossing a River.

current. The velocity of the water

cyclist. Recall that vector addition is commutative.

d. 4r and 4v

directed perpendicular to the

direction under constant acceleration .

displacement and velocity, assuming the acceleration remains the same?

Displacement, Velocity, and Acceleration

divided by the elapsed time.

eliminate the time t and find the

entice the monkey down, the zookeeper points his tranquilizer gun directly at the

equation of the trajectory y as a

Equations of Kinematics

found in part (b), where did he land? 43

from the figure to find:

from the origin to the particle . The displacement of the particle as it moves from

From the top of a cliff, a person throws two stones. The stones have identical initial

From y = (vo sinθo ) t − ½ g t2 and t = 2t1 , one gets:

function of x, v0 and

gets to the monkey before he hits the ground) .

identical initial speeds

If vo sinθo is less than , the projectile will strike the floor before reaching the target ./ 2gd

If we solve for t we can find the trajectory:

in the instantaneous velocity vector, ∆v , to the elapsed time , ∆t :

In the x direction, the spacecraft has an initial velocity

influence of gravity alone, an object near the

into the air and fire it. In the absence of air resistance, where would

is thrown at the same angle above the horizontal. Neglecting air resistance, which

is vertical and positive upward, then under the

It is important to recognize that various changes may

its downward gravitational acceleration.

kinematic variables associated with each direction.

kinematics problem. 14

leaves the gun barrel, intending to land on the ground and escape. Show that the dart

leaves the origin with a velocity v0.

leaves the top of the cliff

meet it. How fast must he run in order to catch the ball just

memorize

memorize

monkey and shoots. The clever monkey lets go of the tree at the same instant the dart

monkey drops straight down, so for him, x = xT at all times. For the dart, x is given by xP=(vocosθo)t . When

motorcycle. (See Figure). The takeoff

moving to the right at constant velocity. You point a rifle straight up

moving to the right at constant velocity. You point a rifle straight up into

of the kick) starts running at the instant the ball is kicked to

of the kinematic variables. Do this for x and y. Select the

of the ratio ∆v / ∆t as ∆t approaches zero :

off a cliff with a running

off the cliff with

one-dimensional free fall, and his height at any time is given by: y=yo+vyot−½ g t2 .The initial height is

P to Q in the time interval ∆t = tf − ti is equal to the vector ∆r =rf − ri .

Q is in the direction of the change in velocity, ∆v = vf − vi

ramp was inclined at

ramp. You can ignore air resistance.

range; other angles fall shorter.

relative to the shore is 2.0 m/s.

relative to the shore?

relative to the water is 4.0 m/s

Rifle is pointed

same range but different in the maximum height and the

So we see that if xT tanθo= (vo sinθo)t at the time when the two x’s are equal, we have a hit. To prove

Solution: To show that the dart hits the monkey, we have to prove that there is some time when the monkey

Special cases for acceleration:

speed be just as she

speed of the ball is 22 m/s. Ignoring air resistance, determine the

speeds, but stone 1 is thrown downward at same angle below the horizontal and stone 2

standing at a distance of 30.0 m from the first (in the direction

stone, if either, strikes the water with greater velocity?

Suppose you are driving a convertible with the top down. The car is

Suppose you are driving a convertible with the top down. The car is

surface of the Earth will accelerate downwards at

that this happens , substitute from the expression of t above. The right hand side becomes :

The “average acceleration” , a , is defined as the the ratio of the change

The “average acceleration” vector a for a particle moving from P to

The “instantaneous acceleration” , a , is defined as the limiting value

the air and fire it. In the absence of air resistance, where would the bullet

the air from the edge of a

The airplane is moving horizontally with a constant velocity of

the bullet land – behind you, ahead of you, or in the barrel of the rifle?

the care package to hit the ground.

The change in the velocity vector is the result of acceleration in the negative

the direction of motion at each instant of time.

The displacement vector r of a projectile

The engine of a boat drives it

The hunter and the monkey: http://www.physicsclassroom.com/mmedia/vectors/mzs.cfm

the initial position of the monkey always hits him, no matter what vo is.

The instantaneous velocity indicates how fast the car moves and

the particle has coordinates (R/2,h).

the projectile if gravity were absent, and the

The river itself was 100 m below the

The vector vot would be the displacement of

The velocity vector v changes with time in both magnitude and direction.

there is ● no acceleration along the horizontal direction.

these are equal , xT = (vocosθo)t , or:

These formulas are useful when θo and vo are given and for a symmetric path .

This is a parabola passing through the origin (completely specified if vo and

time t, its displacement and velocity are r and v. At time 2t, what would be its

top of the ramp for him to have just

various angles of projection.

vector ½ gt2 is its vertical displacement due to

Vector representations and rectangular components of the

Vector representations:

velocity and displacement of a particle moving with a

velocity vo. The maximum height of the projectile is h,

vertical cliff, as shown in

vPG = velocity of the passenger relative to ground,

vPT = velocity of the passenger relative to train,

vTG = velocity of the train relative to ground.

Wagih Ghobriel

was 40.0 m wide, and the far bank was

WC WT TC= +v v v

We conclude that projectile motion is the superposition of

We have proved that at the time the x-coordinates are equal, the y-coordinates are also equal ; a dart aimed at

What are the magnitude and direction of the final velocity of the

What is the time of flight between kickoff and landing?

What is the velocity of the boat

What must her minimum

Where is the other one?

which is 1.75 m wide

whose initial velocity at the origin is vo .

wide. The velocity of the boat

with time .

with time as its magnitude (speed) remains constant .

y direction. The x component of velocity remains constant in time because

yo=(xT tanθo), and we find for the monkey and for the projectile:

yT= xT tanθo− ½ gt2 and yP= (vo sinθo)t− ½ gt2

 Collision will not always happen . There is the further restriction that a collision results only when:

 Note that complementary values of θ o will result in the

 With gravity, both “fall” the same distance ½ gt2 below their g = 0 positions , and the dart still hits

 With no gravity (g = 0) the monkey would remain motionless, and the dart would travel in a straight

Let the effect of air resistance is negligible. If we

 Acceleration parallel or,

 In the horizontal direction, it moves with constant velocity.

 In the vertical direction, a projectile behaves like a freely

 perpendicular to the object’s velocity.