Video Transcript
In this video, we will learn how to
calculate the momentum of a particle moving in a straight line using the formula 𝐇
equals 𝑚𝐯. Imagine two objects, a truck moving
at 30 miles per hour along a road and a paper aeroplane moving at two miles per hour
through the air. Which object would require a
greater force to stop it in the same amount of time?
Intuitively, we know that the truck
would require the greater force to stop it because it has a greater mass and it is
moving faster. We can therefore say that the truck
has a greater momentum. Momentum can be thought of as a
measure of how difficult it is to stop an object that is moving.
Let’s begin by considering a more
formal definition of this. The two factors that contribute to
an object’s momentum are its mass 𝑚 and its velocity vector 𝐯. The greater the mass of the object,
the greater its momentum. And in the same way, the greater
the velocity of the object, the greater its momentum. We can therefore define the
momentum of an object vector 𝐇 as equal to its mass 𝑚 multiplied by its velocity
vector 𝐯. This is written 𝐇 is equal to
𝑚𝐯.
Since velocity is a vector quantity
and mass is a scalar quantity, momentum is a vector quantity. However, we often just want the
magnitude of the momentum. We can therefore write that the
magnitude of vector 𝐇 is equal to the magnitude of 𝑚𝐯. And since mass is a scalar
quantity, we can take it outside of the magnitude sign, giving us the magnitude of
vector 𝐇 is equal to 𝑚 multiplied by the magnitude of vector 𝐯. On the right-hand side, the
magnitude of vector 𝐯 is simply the magnitude of the velocity or the speed. We can denote the magnitude of the
momentum just as 𝐻 and the speed as just 𝑣. This gives us 𝐻 is equal to
𝑚𝑣.
If we consider a bowling ball with
a mass of 12 kilograms moving at a speed of five meters per second along the lane of
a bowling alley, we can calculate the momentum of the bowling ball by substituting
the values into the formula 𝐻 equals 𝑚𝑣. The momentum 𝐻 is equal to 12
kilograms multiplied by five meters per second. As 12 multiplied by five is equal
to 60, the momentum is equal to 60 kilogram meters per second. This shows us that the standard
unit of momentum is kilogram meters per second. However, momentum can also be
measured in other units, in fact any unit of mass multiplied by a unit of speed.
We will now look at some
examples.
Determine the momentum of a car of
mass 2.1 metric tons moving at 42 kilometers per hour.
We begin by recalling that the two
factors that affect an object’s momentum are its mass and speed such that the
momentum 𝐻 is equal to the mass 𝑚 multiplied by the speed 𝑣. In this question, we are told the
mass of the car is 2.1 metric tons and its speed is 42 kilometers per hour. Substituting these values into the
formula, we see that 𝐻 is equal to 2.1 tons multiplied by 42 kilometers per
hour. To calculate 2.1 multiplied by 42,
we can multiply two by 42 and then 0.1 by 42. These are equal to 84 and 4.2,
respectively. As these two values sum to give us
88.2, 2.1 multiplied by 42 is equal to 88.2. We can therefore conclude that the
momentum of the car is 88.2 ton kilometers per hour.
This is a slightly unusual unit as
the standard unit of momentum is kilogram meters per second. However, any unit of mass
multiplied by a unit of speed is a valid unit of momentum. Had we been required to give our
answer in kilogram meters per second, we could’ve converted the mass to kilograms
and the speed to meters per second before multiplying our values.
In our next example, we will
calculate the momentum by first using the equations of motion.
Calculate the momentum of a stone
of mass 520 grams after it has fallen 8.1 meters vertically downward. Consider the acceleration due to
gravity to be 𝑔, which is equal to 9.8 meters per second squared.
In this question, we’re not given
the speed of the stone, which we need to know in order to calculate the
momentum. We are told how far the stone falls
and what its acceleration is. As we are not told anything about
the initial motion of the stone, we can assume that initially it was at rest.
Knowing these three pieces of
information means we can use the equations of motion or SUVAT equations to help
calculate the speed of the particle after it has fallen 8.1 meters. We know that displacement 𝑠 is 8.1
meters. The initial velocity 𝑢 is zero
meters per second. We are trying to calculate the
final velocity 𝑣 in meters per second. And we are told the acceleration
due to gravity is 9.8 meters per second squared.
In this question, we know nothing
about the time. We can therefore use the equation
𝑣 squared is equal to 𝑢 squared plus two 𝑎𝑠. Substituting in our values of 𝑢,
𝑎, and 𝑠, we have 𝑣 squared is equal to zero squared plus two multiplied by 9.8
multiplied by 8.1. 𝑣 squared is therefore equal to
158.76. Square rooting both sides of this
equation and recognizing that the speed must be positive, we have a value of 𝑣
equal to 12.6. After the stone has fallen 8.1
meters vertically downwards, it is traveling at a speed of 12.6 meters per
second.
Next, we recall that the momentum
of any particle 𝐻 is equal to its mass 𝑚 multiplied by its speed 𝑣. The standard units of momentum are
kilogram meters per second, as the standard units of mass are kilograms and speed
are meters per second. We are told that the mass of the
stone is 520 grams, and we know there are 1,000 grams in a kilogram. This means that the mass of the
stone in kilograms is 0.52.
We can therefore calculate the
momentum of the stone by multiplying 0.52 by 12.6. Typing this into the calculator
gives us 6.552. We can therefore conclude that by
the time the stone has fallen a distance of 8.1 meters, it had a momentum of 6.552
kilogram meters per second.
In our next question, we will
calculate the change in momentum of a particle given its acceleration and initial
velocity.
A body of mass 17 kilograms moves
in a straight line with constant acceleration of 1.8 meters per second squared. Its initial velocity is 22.3 meters
per second. Find the increase in its momentum
in the first five seconds.
This question asks us to find the
increase in momentum or change in momentum over a given time period. In order to do this, we need to
work out the difference between its final momentum and its initial momentum. We can express this mathematically
as Δ𝐻 is equal to 𝐻 sub two minus 𝐻 sub one, where Δ𝐻 is the change in momentum,
𝐻 two is the final momentum, and 𝐻 one is the initial momentum. As momentum is equal to mass
multiplied by speed, this can be rewritten as Δ𝐻 is equal to 𝑚𝑣 sub two minus
𝑚𝑣 sub one, where 𝑣 sub two is the final speed, 𝑣 sub one is the initial speed,
and 𝑚 is the mass of the body.
We are told in the question the
initial speed of the body is 22.3 meters per second. We also know that the mass is 17
kilograms. This means we need to begin by
calculating the final speed. We can do this using our equations
of motion, sometimes known as the SUVAT equations.
We know that the initial velocity
is 22.3 meters per second. The acceleration 𝑎 is 1.8 meters
per second squared. And the time period we are
interested in is five seconds. We can therefore calculate the
final velocity 𝑣 using the equation 𝑣 is equal to 𝑢 plus 𝑎𝑡. Substituting in the values of 𝑢,
𝑎, and 𝑡, we have 𝑣 is equal to 22.3 plus 1.8 multiplied by five. 1.8 multiplied by five is equal to
nine. Therefore, 𝑣 is equal to 31.3. The velocity of the body after five
seconds is 31.3 meters per second.
We now have values of 𝑚, 𝑣 sub
one, and 𝑣 sub two. We know the initial speed, final
speed, and the mass of the object. The change in momentum is therefore
equal to 17 multiplied by 31.3 minus 17 multiplied by 22.3. Whilst we could type this straight
into our calculator, we notice that the mass 𝑚 is common to both terms on the
right-hand side. We can therefore rewrite the change
in momentum as 𝑚 multiplied by 𝑣 sub two minus 𝑣 sub one.
In this question, the change in
momentum is equal to 17 multiplied by 31.3 minus 22.3. This simplifies to 17 multiplied by
nine, which in turn is equal to 153. As the mass of the body was
measured in kilograms and the speed or velocities were in meters per second, we use
the standard units of momentum of kilogram meters per second. In the first five seconds of
motion, the momentum of the object increases by 153 kilogram meters per second.
In our final question, we will
calculate the momentum given an equation for the particle’s displacement and the
mass of the particle.
A car of mass 1,350 kilograms moves
in a straight line such that at time 𝑡 seconds, its displacement from a fixed point
on the line is given by 𝑠 is equal to six 𝑡 squared minus three 𝑡 plus four
meters. Find the magnitude of the car’s
momentum at 𝑡 equals three seconds.
In this question, we are given a
function for the position of the car that depends only on time. In order to find the momentum of
the car at a particular time, we are going to need to know its velocity at that
time. In order to get this velocity, we
will firstly need to get a general function for the velocity of the car over
time.
We recall that the velocity of an
object vector 𝐯 is defined as the rate of change of the displacement of the
object. It is therefore the derivative of
the displacement of the object with respect to time. Since this question is just about
an object moving in one dimension, we can use scalar quantities to represent the
velocity 𝑣 and the displacement 𝑠, giving us 𝑣 is equal to d𝑠 by d𝑡. Since 𝑠 is equal to six 𝑡 squared
minus three 𝑡 plus four, we can differentiate this term by term. The velocity 𝑣 is therefore equal
to 12𝑡 minus three. As the displacement was given in
the standard unit of meters, the velocity will be in meters per second.
We want to know the speed of the
car at 𝑡 equals three seconds. Substituting this into our equation
gives us 𝑣 is equal to 12 multiplied by three minus three. The speed of the car after three
seconds is therefore 33 meters per second. We recall that the momentum of any
body can be calculated by multiplying its mass by its speed. This means that the momentum of the
car is equal to 1,350, the mass in kilograms, multiplied by 33, the speed in meters
per second. Typing this into the calculator
gives us 44,550. As the mass is in kilograms and the
speed in meters per second, the momentum will be in its standard units of kilogram
meters per second. At 𝑡 equals three seconds, the car
has a momentum of 44,550 kilogram meters per second.
We will now summarize the key
points from this video. The momentum vector 𝐇 of an object
is the product of its mass 𝑚 and its velocity vector 𝐯 such that 𝐇 is equal to 𝑚
multiplied by 𝐯. Momentum is typically measured in
units of kilogram meters per second. As we have seen in this video,
sometimes we may need to use the kinematic equations or SUVAT equations to find the
velocity of an object in order to calculate its momentum. In our last example, we saw that if
we are given a function for the position of an object at a time 𝑡, we can take the
derivative of that function with respect to time to get a function for the
velocity. This can then be used to help us
calculate the momentum.
