# 4.1 Displacement and velocity vectors  (Page 4/7)

 Page 4 / 7

An example illustrating the independence of vertical and horizontal motions is given by two baseballs. One baseball is dropped from rest. At the same instant, another is thrown horizontally from the same height and it follows a curved path. A stroboscope captures the positions of the balls at fixed time intervals as they fall ( [link] ).

It is remarkable that for each flash of the strobe, the vertical positions of the two balls are the same. This similarity implies vertical motion is independent of whether the ball is moving horizontally. (Assuming no air resistance, the vertical motion of a falling object is influenced by gravity only, not by any horizontal forces.) Careful examination of the ball thrown horizontally shows it travels the same horizontal distance between flashes. This is because there are no additional forces on the ball in the horizontal direction after it is thrown. This result means horizontal velocity is constant and is affected neither by vertical motion nor by gravity (which is vertical). Note this case is true for ideal conditions only. In the real world, air resistance affects the speed of the balls in both directions.

The two-dimensional curved path of the horizontally thrown ball is composed of two independent one-dimensional motions (horizontal and vertical). The key to analyzing such motion, called projectile motion , is to resolve it into motions along perpendicular directions. Resolving two-dimensional motion into perpendicular components is possible because the components are independent.

## Summary

• The position function $\stackrel{\to }{r}\left(t\right)$ gives the position as a function of time of a particle moving in two or three dimensions. Graphically, it is a vector from the origin of a chosen coordinate system to the point where the particle is located at a specific time.
• The displacement vector $\text{Δ}\stackrel{\to }{r}$ gives the shortest distance between any two points on the trajectory of a particle in two or three dimensions.
• Instantaneous velocity gives the speed and direction of a particle at a specific time on its trajectory in two or three dimensions, and is a vector in two and three dimensions.
• The velocity vector is tangent to the trajectory of the particle.
• Displacement $\stackrel{\to }{r}\left(t\right)$ can be written as a vector sum of the one-dimensional displacements $\stackrel{\to }{x}\left(t\right),\stackrel{\to }{y}\left(t\right),\stackrel{\to }{z}\left(t\right)$ along the x , y , and z directions.
• Velocity $\stackrel{\to }{v}\left(t\right)$ can be written as a vector sum of the one-dimensional velocities ${v}_{x}\left(t\right),{v}_{y}\left(t\right),{v}_{z}\left(t\right)$ along the x , y , and z directions.
• Motion in any given direction is independent of motion in a perpendicular direction.

## Conceptual questions

What form does the trajectory of a particle have if the distance from any point A to point B is equal to the magnitude of the displacement from A to B ?

straight line

Give an example of a trajectory in two or three dimensions caused by independent perpendicular motions.

If the instantaneous velocity is zero, what can be said about the slope of the position function?

The slope must be zero because the velocity vector is tangent to the graph of the position function.

## Problems

The coordinates of a particle in a rectangular coordinate system are (1.0, –4.0, 6.0). What is the position vector of the particle?

$\stackrel{\to }{r}=1.0\stackrel{^}{i}-4.0\stackrel{^}{j}+6.0\stackrel{^}{k}$

The position of a particle changes from ${\stackrel{\to }{r}}_{1}=\left(2.0\text{​}\stackrel{^}{i}+3.0\stackrel{^}{j}\right)\text{cm}$ to ${\stackrel{\to }{r}}_{2}=\left(-4.0\stackrel{^}{i}+3.0\stackrel{^}{j}\right)\phantom{\rule{0.2em}{0ex}}\text{cm}.$ What is the particle’s displacement?

The 18th hole at Pebble Beach Golf Course is a dogleg to the left of length 496.0 m. The fairway off the tee is taken to be the x direction. A golfer hits his tee shot a distance of 300.0 m, corresponding to a displacement $\text{Δ}{\stackrel{\to }{r}}_{1}=300.0\phantom{\rule{0.2em}{0ex}}\text{m}\stackrel{^}{i},$ and hits his second shot 189.0 m with a displacement $\text{Δ}{\stackrel{\to }{r}}_{2}=172.0\phantom{\rule{0.2em}{0ex}}\text{m}\stackrel{^}{i}+80.3\phantom{\rule{0.2em}{0ex}}\text{m}\stackrel{^}{j}.$ What is the final displacement of the golf ball from the tee?

$\text{Δ}{\stackrel{\to }{r}}_{\text{Total}}=472.0\phantom{\rule{0.2em}{0ex}}\text{m}\stackrel{^}{i}+80.3\phantom{\rule{0.2em}{0ex}}\text{m}\stackrel{^}{j}$

A bird flies straight northeast a distance of 95.0 km for 3.0 h. With the x -axis due east and the y -axis due north, what is the displacement in unit vector notation for the bird? What is the average velocity for the trip?

A cyclist rides 5.0 km due east, then 10.0 km $20\text{°}$ west of north. From this point she rides 8.0 km due west. What is the final displacement from where the cyclist started?

$\text{Sum of displacements}=-6.4\phantom{\rule{0.2em}{0ex}}\text{km}\stackrel{^}{i}+9.4\phantom{\rule{0.2em}{0ex}}\text{km}\stackrel{^}{j}$

New York Rangers defenseman Daniel Girardi stands at the goal and passes a hockey puck 20 m and $45\text{°}$ from straight down the ice to left wing Chris Kreider waiting at the blue line. Kreider waits for Girardi to reach the blue line and passes the puck directly across the ice to him 10 m away. What is the final displacement of the puck? See the following figure.

The position of a particle is $\stackrel{\to }{r}\left(t\right)=4.0{t}^{2}\stackrel{^}{i}-3.0\stackrel{^}{j}+2.0{t}^{3}\stackrel{^}{k}\text{m}.$ (a) What is the velocity of the particle at 0 s and at $1.0$ s? (b) What is the average velocity between 0 s and $1.0$ s?

a. $\stackrel{\to }{v}\left(t\right)=8.0t\stackrel{^}{i}+6.0{t}^{2}\stackrel{^}{k},\phantom{\rule{0.7em}{0ex}}\stackrel{\to }{v}\left(0\right)=0,\phantom{\rule{0.7em}{0ex}}\stackrel{\to }{v}\left(1.0\right)=8.0\stackrel{^}{i}+6.0\stackrel{^}{k}\text{m/s}$ ,
b. ${\stackrel{\to }{v}}_{\text{avg}}=4.0\text{​}\stackrel{^}{i}+2.0\stackrel{^}{k}\phantom{\rule{0.2em}{0ex}}\text{m/s}$

Clay Matthews, a linebacker for the Green Bay Packers, can reach a speed of 10.0 m/s. At the start of a play, Matthews runs downfield at $45\text{°}$ with respect to the 50-yard line and covers 8.0 m in 1 s. He then runs straight down the field at $90\text{°}$ with respect to the 50-yard line for 12 m, with an elapsed time of 1.2 s. (a) What is Matthews’ final displacement from the start of the play? (b) What is his average velocity?

The F-35B Lighting II is a short-takeoff and vertical landing fighter jet. If it does a vertical takeoff to 20.00-m height above the ground and then follows a flight path angled at $30\text{°}$ with respect to the ground for 20.00 km, what is the final displacement?

$\text{Δ}{\stackrel{\to }{r}}_{1}=20.00\phantom{\rule{0.2em}{0ex}}\text{m}\stackrel{^}{j},\text{Δ}{\stackrel{\to }{r}}_{2}=\left(2.000\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}\text{m}\right)\phantom{\rule{0.2em}{0ex}}\left(\text{cos}30\text{°}\stackrel{^}{i}+\text{sin}\phantom{\rule{0.2em}{0ex}}30\text{°}\stackrel{^}{j}\right)$
$\text{Δ}\stackrel{\to }{r}=1.700\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}\text{m}\stackrel{^}{i}+1.002\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}\text{m}\stackrel{^}{j}$

#### Questions & Answers

Can any one give me the definition for Bending moment plz...
I need a question for moment
what is charge
An attribution of particle that we have thought about to explain certain things like Electomagnetism
Nikunj
please what is the formula instantaneous velocity in projectile motion
A computer is reading from a CD-ROM that rotates at 780 revolutions per minute.What is the centripetal acceleration at a point that is 0.030m from the center of the disc?
change revolution per minute by multiplying from 2pie and devide by 60.and take r=.030 and use formula centripital acceleration =omega sqare r.
Kumar
OK thank you
Rapqueen
observation of body boulded
a gas is compressed to 1/10 0f its original volume.calculate the rise temperature if the original volume is 400k. gamma =1.4
the specific heat of hydrogen at constant pressure and temperature is 14.16kj|k.if 0.8kg of hydrogen is heated from 55 degree Celsius to 80 degree Celsius of a constant pressure. find the external work done .
Celine
hi
shaik
hy
Prasanna
g
what is imaginary mass and how we express is
what is imaginary mass how we express it
Yash
centre of mass is also called as imaginary mass
Lokmani
l'm from Algeria and fell these can help me
Many amusement parks have rides that make vertical loops like the one shown below. For safety, the cars are attached to the rails in such a way that they cannot fall off. If the car goes over the top at just the right speed, gravity alone will supply the centripetal force. What other force acts and what is its direction if: (a) The car goes over the top at faster than this speed? (b) The car goes over the top at slower than this speed?
how can I convert mile to meter per hour
1 mile * 1609m
Boon
hey can someone show me how to solve the - "Hanging from the ceiling over a baby bed ...." question
i wanted to know the steps
Shrushti
sorry shrushti..
Rashid
which question please write it briefly
Asutosh
Olympus Mons on Mars is the largest volcano in the solar system, at a height of 25 km and with a radius of 312 km. If you are standing on the summit, with what initial velocity would you have to fire a projectile from a cannon horizontally to clear the volcano and land on the surface of Mars? Note that Mars has an acceleration of gravity of 3.7 m/s2 .
what is summit
Asutosh
highest point on earth
Ngeh
पृथवी को इसके अक्ष पर कितने कोणीय चाल से घूमाऐ कि भूमधय पे आदमी का भार इसके वासतविक भार से 3/5अधिक हो
best
Murari
At a post office, a parcel that is a 20.0-kg box slides down a ramp inclined at 30.0° 30.0° with the horizontal. The coefficient of kinetic friction between the box and plane is 0.0300. (a) Find the acceleration of the box. (b) Find the velocity of the box as it reaches the end of the plane, if the length of the plane is 2 m and the box starts at rest.
As an IT student must I take physics seriously?
yh
Bernice
hii
Raja
IT came from physics and maths so I don't see why you wouldn't
conditions for pure rolling
Md