Skills
• Basic maths
• Vectors
• Derivatives

## Contents worksheet

By observing how most bodies move, we have found that, in general, they make it in several dimensions: they spin, go up, down, etc.

In this subject, we will study motion in several dimensions: on one hand, we will see motion resulting from rectilinear displacement, on the other, we will study circular motion.

We will use two fundamental mathematical tools: vectors and the derivative.

## Solved exercises worksheet

Test what you have learned in the subject Motion in Two and Three Dimensions with this list of exercises with their respective solutions and classified by sections.

### Introduction to Motion in Several Dimensions

#### What do you know about motion composition?

dificulty

Determine if the following statements are true or false:

a) The circular motion is a rectilinear motion.
b) Non-rectilinear motions can be described by means of rectilinear motions.
c) A soccer head-butt can be studied as the composition of a u.r.m. and a u.a.r.m.

### Horizontal Launch

#### Initial height in horizontal launch

dificulty

At what height, should you place a cannon capable of launching projectiles at an initial velocity of 230 km/h on the horizontal axis if you would like the projectiles to fall at a distance of 250 meters from where they are shot?

#### Tennis and the horizontal launch

dificulty

A tennis ball located 2 m height is hit by a player with his racket. The ball shoots out horizontally with a velocity of 30 m/s. Answer the following questions:

a) How long does the ball takes to hit the ground?
b) What is the angle of the velocity vector with the x-axis at the time the ball reaches the ground?
c) If before being hit, the ball is 5 meters away from the net, how high does the ball pass above the net?

#### Horizontal launch of a rolling ball

dificulty

A golf ball rolls with constant velocity on the surface of a table 2.5 m above the ground. When it reaches the edge, it falls off the table as if horizontally launched so that at 0.4 s it is at a horizontal distance of 1 m from the edge of the table. Determine:

a) What was the constant velocity of the ball while rolling on the table?
b) Would you know how to determine how far horizontally will be the ball when it hits the ground?
c) What is the distance of the ball from the ground at 0.4 s?

#### Initial distance in horizontal launch

dificulty

Quarantine was declared in a passenger cruise ship due to a contagious viral intoxication. To help, the Red Cross sent a helicopter with a box full of drugs. Since the crew of the helicopter could not land on the ship, it is decided to drop the package on a mat on the ship deck.

Assuming that the cruise travels at 72 km/h and that the helicopter travels in the same direction at 108 km/h, at an altitude of 40 m, at what horizontal distance from the ship should the package be dropped? And at what distance if they are traveling toward each other?

### Projectile Motion

#### Flight time, initial velocity and maximum height in parabolic motion

dificulty

A 1.95 m tall shot put athlete put the shot 25 meters away. Knowing that the path begins with an elevation of 40°, calculate:

1. The shot flight time
2. The shot initial velocity
3. The motion maximum height

#### Gooooooooaall !!! Paraaaaboliccccc!

dificulty

Minute 90 of the game... Lopera approaches the ball to make a direct free kick 40 meters from the goal, takes two steps back and kiiicks. The ball takes off at an elevation of 20°... and GOOOOALLL!!! GOOOOOOOALLL!!!! The ball goes in through the top corner at a height of 1.70 m!!!. After hearing this radio broadcast, can you answer the following questions?

a) From Lopera’s kick to scoring the goal, how long did it take? and what was the initial velocity of the ball at the moment of the kick?
b) What is the maximum height reached by the ball?
c) How fast was the ball going when it reached the goal?

#### Launching distance in parabolic motion

dificulty

We have a small device that can launch missiles with a velocity of . Determine how far we must be to hit a target if:

• the height from which the launch occurs is 1.8 m
• the target is at a height of 1.5 m

#### Elevation angle in a parabolic kick

dificulty

Determine the angle in relation to the horizontal, that you must kick a ball to the goal so that it scores barely touching the upper goal post, which is located at a height of 2.45 m, and 9 m away from the starting point. The ball is launched with a velocity of 82 km/h. Notice that the ball must be at the highest point of its trajectory to enter near the top corner of the goal.

### Angular Quantities

#### Converting degrees to radians

dificulty

Converts the following angles from radians to degrees or degrees to radians:

a) 230º
b) 1.34 rad

#### Angular quantities in tractor tires

dificulty

A farmer has a yellow tractor whose front tires have a radius of 40 cm and the rear tires have a radius of 1.20 m. When the front wheels have completed 10 revolutions, how many revolutions will the rear tires have completed?

### Uniform Circular Motion (U.C.M.)

#### Position vector in u.c.m.

dificulty
A body describes a uniform circular motion with a 3 m radius. What is its position vector when its angle is 30º?

#### Radius in a u.c.m.

dificulty

An electric car to scale moves in a circular track describing a uniform circular motion. If the center of the track is placed in the position (0, 0) m determine:

a) The position vector when it is in position (3, 4) m.
b) The radius of the circular trajectory it describes.
c) Its angular position when it is in the position (3, 4) m.

#### Position vector from distance traveled in u.c.m.

dificulty
A body moves describing a circular trajectory with constant linear and angular velocity equal to 3 m/s and 6π rad/s respectively. Determine its position vector when it has traveled exactly one meter.

### Equations of Uniform Circular Motion (U.C.M.)

#### A train ride

dificulty

A toy train nicknamed "torpedo" moves in a circular trajectory of 2 m radius without possibility of changing its linear velocity. Knowing it takes 10 seconds for a full rotation, calculate:

a) Its angular and linear velocities.
b) The angle and the distance traveled in 2 minutes.
c) Its acceleration.

#### Normal acceleration in a wheel

dificulty

A 1 m radius circular disk rotates at constant angular velocity, so that it takes 1.2 s for a full rotation. What is the normal or centripetal acceleration of the external points of its periphery?

#### Centripetal acceleration of the Moon

dificulty

Determine the centripetal acceleration of the Moon knowing that a complete orbit around the Earth takes 27.32 days (sidereal period) and the average distance is 384000 km.

#### An encounter in u.c.m.

dificulty

Two bodies, c1 and c2, begin to move from the same point at constant angular velocity, but in opposite directions, along a 30 m radius circumference. If the first one takes 20 seconds to complete a rotation and the second takes 60 seconds, calculate:

a) The time that they take to meet.
b) The angle and distance traveled by each one.

### Uniform Circular Motion (U.C.M.) Graphs

#### s–t, v–t, at–t and an–t graphs in u.c.m.

dificulty

Based on the graphs of angular position–time (φ–t), angular velocity–time (ω–t), angular acceleration–time (α–t) of uniform circular motion (u.c.m.) draw the graphs of distance traveled–time (s–t), velocity–time (v–t), tangential acceleration–time (at–t) and normal acceleration–time (an–t). Considering that angular velocity is ω>0.

### Equations of Non-Uniform Circular Motion

#### Non-u.c.m. in the blades of a fan

dificulty

A fan whose blades measure 30cm is located in the roof spinning at 140 r.p.m. A blackout makes the fan stops, after 25 seconds. Calculate:

1. Angular acceleration
2. Distance traveled by the end of a blade until it stops, and the number of rotations completed
3. The value of the linear velocity, tangential, normal and full acceleration at 15 seconds after the power outage

## Formulas worksheet

Here is a full formulary by subject Motion in Two and Three Dimensions. By understanding each equation, you will be able to solve any problem that you may find at this level.

Click on the icon    to export them to any external compatible program.

### Horizontal Launch

#### Equation of position in rectilinear uniform motion -x-axis

$x={x}_{0}+v\cdot t$

#### Position equation in free fall

$y=\mathrm{H}-\frac{1}{2}g{t}^{2}$

#### Equation of speed in free fall

$v=-g\cdot t$

#### Equation of acceleration on the Earth surface

$a=-g$

#### Equation of acceleration in uniform rectilinear motion

$a=0$

#### Equation of velocity in uniform rectilinear motion

$v={v}_{0}=\text{cte}$

### Projectile Motion

#### Equation of position x-axis

$x={v}_{x}\cdot t={v}_{0}·\mathrm{cos}\left(\alpha \right)·t$

#### Equation of position y-axis

$y=\mathrm{H}+{\mathrm{v}}_{0\mathrm{y}}·\mathrm{t}-\frac{1}{2}·g·{t}^{2}=\mathrm{H}+{\mathrm{v}}_{0}·\mathrm{sin}\left(\mathrm{\alpha }\right)·\mathrm{t}-\frac{1}{2}·g·{t}^{2}$

#### Equation of velocity x-axis

${v}_{x}={v}_{0x}={v}_{0}·\mathrm{cos}\left(\alpha \right)$

#### Equation of velocity y-axis

${v}_{y}={v}_{0y}-g\cdot t={v}_{0}·\mathrm{sin}\left(\alpha \right)-g\cdot t$

#### Equation of acceleration in the x-axis

${a}_{x}=0$

#### Equation of acceleration in the y-axis

${a}_{y}=-g$

### Angular Quantities

#### Linear space - angular space relationship

$s=\phi ·R$

#### Average angular velocity

${\omega }_{a}=\frac{∆\phi }{∆t}=\frac{{\phi }_{f}-{\phi }_{i}}{{t}_{f}-{t}_{i}}$

#### Angular velocity

$\omega =\underset{∆t\to 0}{\mathrm{lim}}\frac{∆\phi }{∆t}=\frac{d\phi }{dt}$

#### Average angular acceleration

${\alpha }_{a}=\frac{∆\omega }{∆t}=\frac{{\omega }_{f}-{\omega }_{i}}{{t}_{f}-{t}_{i}}$

#### Angular acceleration

$\alpha =\underset{∆t\to 0}{\mathrm{lim}}\frac{∆\omega }{∆t}=\frac{d\omega }{dt}$

#### Normal acceleration equation (u.c.m. and u.a.c.m.)

${a}_{n}=\frac{{v}^{2}}{R}={\omega }^{2}·R$

#### Tangential acceleration - angular acceleration relationship

${a}_{t}=\alpha ·R$

#### Linear velocity - angular velocity relationship

$v=\omega ·R$

### Uniform Circular Motion (U.C.M.)

#### Position vector in circular motion

$\stackrel{\to }{r}=x·\stackrel{\to }{i}+y·\stackrel{\to }{j}=R·\mathrm{cos}\left(\phi \right)·\stackrel{\to }{i}+R·\mathrm{sin}\left(\phi \right)·\stackrel{\to }{j}$

#### Equation of tangential acceleration (u.c.m.)

${a}_{t}=0$

#### Angular velocity in u.c.m.

$\omega =\text{constant}$

#### Angular acceleration (u.c.m.)

$\alpha =0$

### Equations of Uniform Circular Motion (U.C.M.)

#### Angular position in u.c.m.

$\phi ={\phi }_{0}+\omega ·t$

#### Angular velocity in u.c.m.

$\omega =\text{constant}$

#### Angular acceleration (u.c.m.)

$\alpha =0$

#### Frequency in uniform circular motion (u.c.m.)

$f=\frac{\omega }{2·\pi }$

#### Relationship angular velocity - period - frequency in u.c.m.

$\omega =\frac{2·\pi }{T}=2·\pi ·f$

### Uniform Circular Motion (U.C.M.) Graphs

#### Angular position in u.c.m.

$\phi ={\phi }_{0}+\omega ·t$

#### Angular velocity in u.c.m.

$\omega =\text{constant}$

#### Angular acceleration (u.c.m.)

$\alpha =0$

### Non-Uniform Circular Motion

#### Position vector in circular motion

$\stackrel{\to }{r}=x·\stackrel{\to }{i}+y·\stackrel{\to }{j}=R·\mathrm{cos}\left(\phi \right)·\stackrel{\to }{i}+R·\mathrm{sin}\left(\phi \right)·\stackrel{\to }{j}$

#### Angular position in non-u.c.m.

$\phi ={\phi }_{0}+\omega ·t+\frac{1}{2}·\alpha ·{t}^{2}$

#### Angular velocity in non-u.c.m.

$\omega ={\omega }_{0}+\alpha ·t$

#### Angular acceleration in non-u.c.m.

$\alpha =\text{constant}$

### Equations of Non-Uniform Circular Motion

#### Angular position in non-u.c.m.

$\phi ={\phi }_{0}+\omega ·t+\frac{1}{2}·\alpha ·{t}^{2}$

#### Angular velocity in non-u.c.m.

$\omega ={\omega }_{0}+\alpha ·t$

#### Angular acceleration in non-u.c.m.

$\alpha =\text{constant}$

### Non-Uniform Circular Motion Graphs

#### Angular position in non-u.c.m.

$\phi ={\phi }_{0}+\omega ·t+\frac{1}{2}·\alpha ·{t}^{2}$

#### Angular velocity in non-u.c.m.

$\omega ={\omega }_{0}+\alpha ·t$

#### Angular acceleration in non-u.c.m.

$\alpha =\text{constant}$