Skills
• Basic maths
• Vectors
• Derivatives

Contents worksheet

Motion is one the most evident physical phenomena because it is easily observable. Its study allows us to understand the motion of objects that surely are familiar to you, such as trains, cars and airplanes. But it also provides the basis to study other less common objects, such as satellites, planets, stars and many more.

The branch of Physics responsible for the study of this phenomenon is Kinematics, which study the laws of motion without considering the causes of such motion.

In this section, we will present these laws using two fundamental mathematical tools: vectors and derivatives.

Solved exercises worksheet

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

Motions and Reference Frames

Reference Frames in Kinematics

dificulty

Say whether the following statements are true or false:

a) Reference frames can be classified as inertial and heterogeneous.
b) The study of motion is independent of the observer´s position in relation to the motion.

Position Vector

What is the position vector?

dificulty

Find the position vector and its magnitude for the following points:

a) P1 (-1,2)
b) P2 (-3,4)

Trajectory and Equation of Position

Position Equation of the S.H.S. Madrid-Barcelona (Spain High Speed train)

dificulty

The S.H.S. Madrid-Barcelona during the first stage of its trip travels east at a rate of 50 m each second and North at 30 m each second. Calculate:

1. The motion x-coordinate as function of time
2. The motion y-coordinate as a function of time
3. The position equation of the train for the studied stretch
4. The distance to the origin of the section as a function of time

Distance Traveled

Crescent Shaped Motion

dificulty

A body moves between any two instants of time following a circular path with a radius of 5 meter, as you can see in the figure.

Determine:

1. The displacement vector of the body and the distance traveled, assuming the origin of the system of reference is located at the starting point of the motion
2. The displacement vector of the body and the distance traveled, assuming the origin of the system of reference is in the center of the semicircle

Difference between Displacement and Distance Traveled

What do you know about displacement and distance traveled?

dificulty

Answer if the following statements are true or false:

a) Distance traveled only depends on the starting and final positions without consideration of the trajectory.
b) Distance traveled is always greater or equal than the magnitude of the displacement vector.
c) Distance traveled it is a vector.
d) As time passes, a body in motion always increases its displacement.

Average Velocity

Calculation of average velocity

dificulty

If a body is in the position (1,2) and after 2 seconds it is in the position (1,-2).

What will be its average velocity considering that all units are expressed in the International System?

Instantaneous Velocity

Velocity starting with the position equation

dificulty

If a body moves according to the following equation:

Calculate its instantaneous velocity at time t=1 s.

A particle in motion

dificulty

The position of a certain particle depends on time according to the equation $x\left(t\right)={t}^{2}-5t+1.2$ , where x is expressed in meters and t in seconds. You are asked to:

1. Determine the displacement and the average velocity during the interval 3.0s ≤ t ≤ 4.0 s.
2. Find the general formula for the displacement for the interval between t and t + Δt.
3. Find the instantaneous velocity for any time t considering the limit whenΔt approaches 0.

Average Speed

Average speed of the Moon

dificulty

Knowing that the Moon takes 28 days to make one complete revolution around the Earth, calculate its average speed.

Datum: Consider the trajectory of the Moon as circular with a radius of 384,000 Km.

Average speed in a circular trajectory

dificulty

A body is in a circular track like the one in the figure.

If the radius is 80 m, calculate:

1. The average speed between points A and B, if the body takes 13 seconds to go from one point to the other
2. The average speed between points A and C if a cyclist takes 26 seconds to go from one point to the other
3. The average speed between point A and C if the rider takes 13 seconds to go from point A to point B

Average speed in a full circle

dificulty

A body does a full circle of 4-meter radius in 1.5 seconds. Calculate:

a) the average speed.
b) the average velocity.

Instantaneous Speed

Speed as a function of distance traveled

dificulty

Knowing that the distance a body travels as a function of time is given by the following equation:

$S\left(t\right)={t}^{2}+5·t+1$

Calculate:

a) The average speed during the first 3 seconds.
b) The instantaneous speed of the body.

Concept of Acceleration

Acceleration graphs

dificulty

Given the following graphs that show the variation of the speed with time of three different bodies, answer the following questions:

a) Which case has the fastest variation of the speed?
b) What does the slope represent in each one of the 3 cases?
c) Why is the line in case I horizontal?
d) In case I, does the possibility exist that the body is experiencing acceleration?

Average Acceleration

dificulty

A basketball player throws the ball with a velocity of , with such bad luck that it bounces off the board with a velocity .

Calculate the average acceleration knowing that the impact against the board lasts exactly 0.02 seconds

Instantaneous Acceleration

Acceleration of a meteorite

dificulty

A meteorite moves through space with a velocity v⃗(t) = (1+4·t) i⃗+t2 j⃗ m. Calculate

a) Its average acceleration between the times t1=2 s and t2= 4 s.
b) Its acceleration at t3= 6 s.

Tangential Acceleration

Magnitude of the tangential acceleration

dificulty

Knowing that the magnitude of the velocity of a body in S.I. units is:

$v=7+2·t+3·{t}^{2}$

Calculate the magnitude of the tangential acceleration.

Types of Motions According to the Acceleration

Identifying motion

dificulty

Answer if the following statements are true or false:

a) If motion has normal acceleration, the motion is rectilinear.
b) If motion has tangential acceleration, the motion is curvilinear.
c) Normal and tangential acceleration are called intrinsic components of the acceleration.
d) Motion without normal acceleration and with constant tangential acceleration is called uniformly accelerated rectilinear motion. (u.a.r.m.)

Rectilinear Motion: Sign Convention

Conversion of vectors to scalars in rectilinear motion

dificulty

Assuming that the following vector magnitudes refer to rectilinear motion, give their corresponding scalar representation:

•
• $∆\stackrel{\to }{r}=-3·\stackrel{\to }{j}m$
•
• $\stackrel{\to }{v}=3·t·\stackrel{\to }{i}m/s$

Equations of Constant Velocity Motion

u.r.m. in collision marbles

dificulty

Two marble players face each other with their marbles in hand. The game consists of throwing the marbles at the same time in a straight line so they hit each other. The players are located 36 meters from each other and player A launches its marble at 2 m/s and player B at 4 m/s, in a uniform rectilinear motion. Calculate that distance from player B at which the marbles will collide.

Time at which two bodies with u.r.m. will meet

dificulty

Two bodies depart from the same point in the same direction, with uniform rectilinear motion. Knowing that they depart 15 seconds apart, that the first one does it at a speed of 20 m/s and the second one at a speed of 24 m/s, determine at which time they will meet and how far from the origin.

u.r.m. velocity in the collision of two bodies

dificulty

Two bodies depart with 15 s of difference from the same point in the same direction. Knowing that the speed of the first one is 72 km/h, what should be the speed of the second to reach it in 90 s?

Note: You can assume that both bodies move with uniform rectilinear motion.

Constant Velocity Motion Graphs

Draw graphs from u.r.m. equations

dificulty

Determine the graphs of the following uniform rectilinear motions:

1. x = 3 + 4·t
2. x = 3 - 4·t
3. x = -3 + 4·t
4. x = -3 - 4·t
5. 3·x = 9 + 12·t

Where x is measured in meters and t in seconds.

x-t graph of a person walking

dificulty
A person walks in a straight line at a velocity of 5 km per hour for 15 minutes. Could you determine its graph of position with respect to time?

Analysis of uniform rectilinear motion graphs

dificulty

The following graph of position-time ( x-t ) corresponds to a body moving in a straight trajectory. Determine the equation of motion for each segment and its velocity-time graph. From the expression for each segment, search for a general expression in the form of a function defined by segments, called  a piecewise-defined function, for the position and one for the velocity.

Equations of Constant Acceleration Motion

A bicycle with constant acceleration

dificulty

A cyclist starts his morning ride and after 10 seconds his velocity is 7.2 km/h. At that moment, he sees a dog approaching and slows down for 6 seconds until the bicycle stops. Calculate:

a) The acceleration until he begins to slow down.
b) The braking acceleration of the bike.
c) The total distance traveled.

Free Fall

Gravity on the Moon

dificulty

Determine the acceleration of gravity on the Moon knowing that if you drop an object from a height of 5 m, it takes 2.47 s to reach the ground.

Glass in free fall

dificulty

A glass of water on the edge of a table falls towards the floor from a height of 1.5 m. Considering that gravity is 10 m/s2, calculate:

a) The time the glass is in the air.
b) The velocity with which it impacts on the ground.

Free fall in a well

dificulty

Determine the depth of a well in which a stone is dropped, and where you can hear the impact on the water after 1.5 s, considering that the velocity of sound is 340 m/s.

Drops in fall free

dificulty

A broken faucet lets out water drops every 1/4 of a second. If the faucet is just 3 meters above the ground, and if one drop falls in this very instant, determine what is the position of the drops still in the air at this moment.

Free fall from unknown height

dificulty
A stone is dropped from a certain height. What will be the velocity of the stone after 8 seconds from the start of its fall if it never gets to touch the ground? What will the distance the stone traveled in that time?

Vertical Launch

A rookie equilibrist

dificulty

A rookie equilibrist is standing on a platform 12 meters above the ground. While practicing juggling with 2 balls, he stumbles and throws both balls at 9 m/s, however, he throws one up which we will call A and the other one down which we will call B. Considering that gravity is 10 m/s2, calculate

a) The time they are in the air.
b) Their velocity when they hit the ground.
c) The maximum height that ball A reaches.

A launch of negligible mass

dificulty

From a height of 40 meters an object of negligible mass is thrown downward with a velocity of 20 m/s. How long will it take to hit the ground? What will be its velocity on impact?

Vertical launch and free fall

dificulty

A stone is let to free fall to the bottom of a cliff with a height of 80 m. A second later a second stone is thrown downward so that it reaches the bottom at the same time as the first one

1. What was the launch velocity of the second stone?
2. What was the velocity of the first stone when they both hit the bottom?
3. How long was the second stone in the air?

Formulas worksheet

Here is a full formulary by subject Motion in Physics. 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.

Position Vector

Modulus position vector in Cartesian coordinates in 3 dimensions

$\left|\stackrel{\to }{r}\right|=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$

Modulus position vector in Cartesian coordinates in 2 dimensions

$\left|\stackrel{\to }{r}\right|=\sqrt{{x}^{2}+{y}^{2}}$

Position vector in 2 dimensional Cartesian coordinates

$\stackrel{\to }{r}=x\stackrel{\to }{i}+y\stackrel{\to }{j}$

Position vector in 3 dimensional Cartesian coordinates

$\stackrel{\to }{r}=x\stackrel{\to }{i}+y\stackrel{\to }{j}+z\stackrel{\to }{k}$

Trajectory and Equation of Position

Position equation in two dimensional Cartesian

$\stackrel{\to }{r}\left(t\right)=x\left(t\right)\stackrel{\to }{i}+y\left(t\right)\stackrel{\to }{j}$

Position equation in three dimensional Cartesian

$\stackrel{\to }{r}\left(t\right)=x\left(t\right)\stackrel{\to }{i}+y\left(t\right)\stackrel{\to }{j}+z\left(t\right)\stackrel{\to }{k}$

Displacement

Displacement vector in three dimensions, Cartesian coordinates

$∆\stackrel{\to }{r}={\stackrel{\to }{r}}_{f}-{\stackrel{\to }{r}}_{i}=\left({x}_{f}-{x}_{i}\right)\stackrel{\to }{i}+\left({y}_{f}-{y}_{i}\right)\stackrel{\to }{j}+\left({z}_{f}-{z}_{i}\right)\stackrel{\to }{k}$

Magnitude of the displacement vector in two-dimensional Cartesian

$\left|∆\stackrel{\to }{r}\right|=\sqrt{{\left({x}_{f}-{x}_{i}\right)}^{2}+{\left({y}_{f}-{y}_{i}\right)}^{2}}$

Displacement vector in two dimensions, Cartesian coordinates

$∆\stackrel{\to }{r}={\stackrel{\to }{r}}_{f}-{\stackrel{\to }{r}}_{i}=\left({x}_{f}-{x}_{i}\right)\stackrel{\to }{i}+\left({y}_{f}-{y}_{i}\right)\stackrel{\to }{j}$

Magnitude of the displacement vector in three-dimensional Cartesian

$\left|∆\stackrel{\to }{r}\right|=\sqrt{{\left({x}_{f}-{x}_{i}\right)}^{2}+{\left({y}_{f}-{y}_{i}\right)}^{2}+{\left({z}_{f}-{z}_{i}\right)}^{2}}$

Distance Traveled

Circumference arc length

$L=\theta \cdot r$

Instantaneous Velocity

Magnitude velocity 3 Cartesian dimensions

$\left|\stackrel{\to }{v}\right|=\sqrt{{v}_{x}^{2}+{v}_{y}^{2}+{v}_{z}^{2}}$

Velocity magnitude 2 Cartesian dimensions

$\left|\stackrel{\to }{v}\right|=\sqrt{{v}_{x}^{2}+{v}_{y}^{2}}$

Instantaneous Speed

Instantaneous speed

$V =\underset{∆t\to 0}{\mathrm{lim}}\frac{∆s}{∆t}$

Average Acceleration

Average Acceleration

${\stackrel{\to }{a}}_{a}=\frac{{\stackrel{\to }{v}}_{2}-{\stackrel{\to }{v}}_{1}}{{t}_{2}-{t}_{1}}=\frac{∆\stackrel{\to }{v}}{∆t}$

Instantaneous Acceleration

Acceleration magnitude in 3 Cartesian dimensions

$\left|\stackrel{\to }{a}\right|=\sqrt{{a}_{x}^{2}+{a}_{y}^{2}+{a}_{z}^{2}}$

Acceleration magnitude in 2 Cartesian dimensions

$\left|\stackrel{\to }{a}\right|=\sqrt{{a}_{x}^{2}+{a}_{y}^{2}}$

Intrinsic Components of Acceleration

Magnitude of the acceleration as a function of intrinsic components

$\left|\stackrel{\to }{a}\right|=\sqrt{{a}_{t}^{2}+{a}_{n}^{2}}$

Acceleration as a function of the intrinsic components

$\stackrel{\to }{a}={\stackrel{\to }{a}}_{t}+{\stackrel{\to }{a}}_{n}={a}_{t}{\stackrel{\to }{u}}_{t}+{a}_{n}{\stackrel{\to }{u}}_{n}$

Tangential Acceleration

Tangential acceleration

${\stackrel{\to }{a}}_{t}=\frac{dv}{dt}{\stackrel{\to }{u}}_{t}$

Normal or Centripetal Acceleration

Normal or centripetal acceleration

${\stackrel{\to }{a}}_{n}=\frac{{v}^{2}}{\rho }{\stackrel{\to }{u}}_{n}$

Equations of Constant Velocity Motion

Equation of position in rectilinear uniform motion -x-axis

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

Equation of acceleration in uniform rectilinear motion

$a=0$

Equation of velocity in uniform rectilinear motion

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

Constant Velocity Motion Graphs

Equation of position in rectilinear uniform motion -x-axis

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

Equation of acceleration in uniform rectilinear motion

$a=0$

Equation of velocity in uniform rectilinear motion

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

Equations of Constant Acceleration Motion

Position equation of uniformly accelerated rectilinear motion - x-axis

$x={x}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$

Position equation of uniformly accelerated rectilinear motion - y-axis

$y={y}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$

Acceleration equation in uniformly accelerated rectilinear motion

$a=\text{cte}$

Velocity equation in uniformly accelerated rectilinear motion

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

Constant Acceleration Motion Graphs

Position equation of uniformly accelerated rectilinear motion - x-axis

$x={x}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$

Velocity equation in uniformly accelerated rectilinear motion

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

Acceleration equation in uniformly accelerated rectilinear motion

$a=\text{cte}$

Free Fall

Position equation in free fall

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

Position equation of uniformly accelerated rectilinear motion - y-axis

$y={y}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$

Equation of speed in free fall

$v=-g\cdot t$

Equation of acceleration on the Earth surface

$a=-g$

Vertical Launch

Equation of position of downward vertical launch

$y=\mathrm{H}-{v}_{0}t-\frac{1}{2}g{t}^{2}$

Equation of position in upward vertical launch

$y=\mathrm{H}+{v}_{0}t-\frac{1}{2}g{t}^{2}$

Position equation of uniformly accelerated rectilinear motion - y-axis

$y={y}_{0}+{v}_{0}t+\frac{1}{2}a{t}^{2}$

Equation of velocity of downward vertical launch vertical

$v=-{v}_{0}-g\cdot t$

Equation of velocity of the upward vertical launch

$v={v}_{0}-g\cdot t$

Equation of acceleration on the Earth surface

$a=-g$