In this video, we will learn how to
solve problems involving motion of a particle with uniform acceleration through one
or more sections of its path. We will begin by recalling the
equations of motion and how we can use them to calculate unknowns.
The equations of motion, also known
as SUVAT equations, are used when acceleration 𝑎 is constant. They are known as SUVAT equations
because they contain the following variables: 𝑠 for displacement, 𝑢 is the initial
velocity, 𝑣 is the final velocity or velocity at time 𝑡, 𝑎 is the acceleration,
and 𝑡 is the time. The initial velocity 𝑢 is also
sometimes written as 𝑣 sub zero. For the purposes of this video, we
will use the letter 𝑢. However, these letters are
Before using the equations, it is
important that our measurements are in the correct units. For displacement, we use meters,
for the velocities meters per second, acceleration is measured in meters per second
squared, and time in seconds. Depending on which variables we’re
dealing with, we use one of the five equations. 𝑣 is equal to 𝑢 plus 𝑎𝑡. 𝑣 squared is equal to 𝑢 squared
plus two 𝑎𝑠. 𝑠 is equal to 𝑢𝑡 plus a half
𝑎𝑡 squared. 𝑠 is equal to 𝑣𝑡 minus a half
𝑎𝑡 squared. And 𝑠 is equal to 𝑢 plus 𝑣
divided by two all multiplied by 𝑡.
Before we look at some specific
questions, we will also look at how we can solve this type of problem using a
velocity–time graph. A velocity–time graph has time in
seconds going along the horizontal or 𝑥-axis and velocity in meters per second
going up the 𝑦- or vertical axis. When dealing with constant
acceleration, the velocity–time graph will be made up of straight lines.
These straight lines can have
positive or negative slope or be horizontal. When the graph is horizontal, the
acceleration is equal to zero meters per second squared. When the graph is sloping
diagonally upwards or downwards, we can calculate the acceleration by dividing the
change in velocity by the change in time. This is the same as calculating the
slope or the gradient. If the slope is positive, the
acceleration will be positive. And if the slope is negative, the
acceleration will be negative. This is also known as a
deceleration. The greater the slope of the graph,
the greater the acceleration or deceleration. We can also think of the change of
velocity over the change in time as being the rise over the run.
We can also calculate the
displacement of the object from the graph. It is the area trapped between the
graph and the 𝑥-axis. We can sometimes calculate this in
one go. In the diagram shown, we would
calculate the area of the trapezoid or trapezium. Alternatively, we could split the
area into triangles and rectangles and calculate each area separately. The displacement, and in this case,
also the distance travelled, is equal to the sum of these areas. Whilst the velocity–time graph can
be used to solve some of the problems that follow, we will focus in this video on
using the equations of motion.
A cyclist riding down a hill
from rest was accelerating at a rate of 0.5 meters per square second. By the time he reached the
bottom of the hill, he was traveling at 1.5 meters per second. He continued traveling at this
speed for another 9.5 seconds. Determine the total distance 𝑠
that the cyclist covered.
There are two parts to the
cyclist’s journey that we need to consider, firstly when he was accelerating
down the hill and secondly when he continued to travel at this constant
speed. We know that when traveling at
a constant speed, acceleration is equal to zero. In order to answer this
question, we will use the equations of motion or SUVAT equations. Let’s begin by considering the
first part of the journey.
The cyclist accelerates from
rest. Therefore, his initial velocity
is zero meters per second. We are told that the
acceleration is 0.5 meters per second squared. At the bottom of the hill, the
cyclist has reached a speed of 1.5 meters per second. Therefore, his final velocity
𝑣 for this part of the journey is 1.5 meters per second. We are trying to calculate the
total distance traveled. Therefore, we will call the
displacement or distance for the first part of the journey 𝑠 one.
We will now use the equation
that 𝑣 squared is equal to 𝑢 squared plus two 𝑎𝑠. Substituting in our values
gives us 1.5 squared is equal to zero squared plus two multiplied by 0.5
multiplied by 𝑠 sub one. This simplifies to 2.25 is
equal to one multiplied by 𝑠 sub one. The distance traveled in the
first part of the journey is, therefore, equal to 2.25 meters.
Let’s now consider the second
part of the journey. The cyclist is traveling at a
constant speed. Therefore, the acceleration is
zero meters per second squared. Both the initial and final
velocities are equal to 1.5 meters per second. And we are told he cycles at
this speed for 9.5 seconds. We will call the distance
covered in this part of the journey 𝑠 sub two. We know that the displacement,
or in this case the distance 𝑠, is equal to 𝑢 plus 𝑣 divided by two
multiplied by 𝑡. Substituting in our values, we
have 𝑠 two is equal to 1.5 plus 1.5 divided by two multiplied by 9.5. This simplifies to 1.5
multiplied by 9.5, which is equal to 14.25. The distance covered in the
second part of the journey is 14.25 meters.
We can now calculate the total
distance covered by adding 2.25 and 14.25. This gives us an answer of 16.5
meters. The cyclist covers a total
distance of 16.5 meters.
In our next question, we’ll
consider the motion of a train between two stations.
A train, starting from rest,
began moving in a straight line between two stations. For the first 80 seconds, it
moved with a constant acceleration 𝑎. Then it continued to move at
the velocity it had acquired for a further 65 seconds. Finally, it decelerated with a
rate of two 𝑎 until it came to rest. Given that the distance between
the two stations was 8.9 kilometers, find the magnitude of 𝑎 and the velocity
𝑣 at which it moved during the middle leg of the journey.
In order to answer this
question, we will use our equations of motion or SUVAT equations. We will create equations for
the three legs of the journey and then solve them to calculate any unknowns. Let’s begin by considering the
first part of the journey. The train accelerates from
rest, so our initial velocity is equal to zero meters per second. It accelerates for 80 seconds
at a constant acceleration of 𝑎 meters per second squared. We will call the velocity it
reaches at this point 𝑣 and the displacement from the start point or the
distance covered 𝑠 one.
We know that 𝑣 is equal to 𝑢
plus 𝑎𝑡. Substituting in our values, we
have 𝑣 is equal to zero plus 𝑎 multiplied by 80. This gives us the equation 𝑣
is equal to 80𝑎. Another one of our equations
states that 𝑠 is equal to 𝑢 plus 𝑣 divided by two multiplied by 𝑡. 𝑠 one is, therefore, equal to
zero plus 𝑣 divided by two multiplied by 80. This simplifies to 𝑠 one is
equal to 40𝑣. We will call these equations
one and two and move on to the second part of the journey.
In the second part of the
journey, the train travels at this constant velocity 𝑣. This means its acceleration is
equal to zero meters per second squared. The time it takes is 65
seconds. And we will call the distance
traveled 𝑠 two. Once again, we will use the
equation 𝑠 is equal to 𝑢 plus 𝑣 divided by two multiplied by 𝑡. This gives us 𝑠 two is equal
to 𝑣 plus 𝑣 divided by two multiplied by 65. As 𝑣 plus 𝑣 is equal to two
𝑣, this simplifies to 𝑠 two is equal to 65𝑣. We will call this equation
three and now move on to the final part of the journey.
In this part of the journey,
the train decelerates to rest. As the deceleration is equal to
two 𝑎, our value of 𝑎 will be equal to negative two 𝑎 meters per second
squared. The final velocity 𝑣 is equal
to zero meters per second. The initial velocity in this
part of the journey is equal to 𝑣, the speed it was traveling for the second
part of the journey. We will call the distance
traveled in this part 𝑠 three and the time it takes 𝑡. We will begin by once again
using 𝑠 is equal to 𝑢 plus 𝑣 over two multiplied by 𝑡. 𝑠 three is, therefore, equal
to 𝑣 plus zero divided by two multiplied by 𝑡. This simplifies to 𝑠 three is
equal to 𝑣𝑡 divided by two.
We will also use the equation
𝑣 equals 𝑢 plus 𝑎𝑡. Substituting in our values here
gives us zero is equal to 𝑣 plus negative two 𝑎 multiplied by 𝑡. This simplifies to zero is
equal to 𝑣 minus two 𝑎𝑡. And adding two 𝑎𝑡 to both
sides of this equation gives us 𝑣 is equal to two 𝑎𝑡. We now have two further
equations that we’ll number four and five.
The final part of our question
tells us that the distance between the two stations was 8.9 kilometers. To use the equations of motion,
we need this to be in meters. As there are 1000 meters in one
kilometer, 8.9 kilometers is equal to 8900 meters. The distances 𝑠 one, 𝑠 two,
and 𝑠 three must, therefore, sum to 8900. Using equations two, three, and
four, this can be rewritten as 40𝑣 plus 65𝑣 plus 𝑣𝑡 over two is equal to
8900. 105𝑣 plus 𝑣𝑡 over two is
equal to 8900. If we can work out the value of
the time 𝑡, we can then calculate the velocity 𝑣.
Let’s consider equation one and
equation five. These tell us that 𝑣 is equal
to 80𝑎 and two 𝑎𝑡, which means that 80𝑎 must be equal to two 𝑎𝑡. At this stage, we know that 𝑎
is not equal to zero. Therefore, we can divide
through by 𝑎. We can then divide both sides
of this equation by two, giving us 𝑡 is equal to 40. The time taken for the third
part of the journey is 40 seconds. We can now substitute 𝑡 equals
40 to calculate 𝑣. 40 divided by two is equal to
20, and 105 plus 20 is 125. Therefore, 125𝑣 is equal to
8900. Dividing both sides of this
equation by 125 gives us 𝑣 is equal to 71.2. The velocity 𝑣 is equal to
71.2 meters per second.
As we now know the value of 𝑣,
we can use equation one to calculate the acceleration 𝑎. 71.2 is equal to 80𝑎. Dividing both sides of this
equation by 80 gives us 𝑎 is equal to 0.89. The magnitude of acceleration
𝑎 is equal to 0.89 meters per second squared. And the velocity 𝑣 is 71.2
meters per second.
In our final question, we will
study the motion of two bullets fired at two different wooden blocks.
A bullet was fired horizontally
at a wooden block. It entered the block at 80
meters per second and penetrated 32 centimeters into the block before it
stopped. Assuming that its acceleration
𝑎 was uniform, find 𝑎. If under similar conditions,
another bullet was fired at a wooden block that was 14 centimeters thick,
determine the velocity at which the bullet exited the wooden block.
In our first scenario, we are
told that a bullet is fired with velocity 80 meters per second. It penetrates 32 centimeters
into the block. And as there are 100
centimeters in one meter, this is equal to 0.32 meters. To calculate the value of 𝑎,
we will use the equations of uniform acceleration or SUVAT equations. We know that 𝑠, the
displacement or distance, is equal to 0.32 meters, the initial velocity is 80
meters per second, the final velocity is zero meters per second, and we are
trying to calculate the acceleration 𝑎. We will do this using the
equation 𝑣 squared is equal to 𝑢 squared plus two 𝑎𝑠.
Substituting in our values
gives us zero squared is equal to 80 squared plus two 𝑎 multiplied by 0.32. This simplifies to zero is
equal to 6400 plus 0.64𝑎. We can then subtract 6400 from
both sides and then divide by 0.64. This gives us a value of 𝑎
equal to negative 10000. The acceleration of the bullet
is negative 10000 meters per second squared. As there are 1000 meters in a
kilometer, this can also be written as negative 10 kilometers per second
In our second scenario, the
bullet travels straight through a wooden block of thickness 14 centimeters or
0.14 meters. Using the same values of 𝑢 and
𝑎, we can now calculate 𝑣, the velocity, at which the bullet exits the wooden
block. We know that 𝑣 squared is
equal to 𝑢 squared plus two 𝑎𝑠. Substituting in our values, we
can calculate 𝑣 squared. This is equal to 3600. Square rooting both sides and
knowing that 𝑣 must be positive, we get 𝑣 is equal to 60. The bullet exits the block at a
velocity of 60 meters per second.
We will now summarize the key
points from this video. The equations of motion can be used
when the acceleration of an object is constant. Before using any of the five
equations, it is important we’re in the correct units of measurement. Our standard units are meters,
seconds, meters per second, and meters per second squared.