SPU4U1- TRAJECTORIES AND IT’S
HORIZONTAL AND VERTICAL DISPLACEMENT
LAB REPORT
This lab report is presented to Dr.Cramer by Abdullah Mustafa. The experiment
was conducted on Wednesday 21st, September 2022 in room 422. The group of
students conducting the experiment include me, Arash and Huzaifa.
INTRODUCTION AND THEORY
The purpose of this lab is studying the horizontal and vertical displacement of an
object experiencing projectile motion and the relation between them.For this
purpose, we will use a steel ball and measure its horizontal and vertical
displacement as it falls off the ramp and use the data to see what is the relation
between its horizontal and vertical displacement. We also will be linearizing the
graph between the components.
Projectile motion is when an object is projected near the Earth’s surface and it
moves along a curved path due to gravity . The displacement ( Δd) of the object
can be broken down into its horizontal and vertical components (Δd
x and
Δdy). The purpose of this lab is to find the relation between the vertical and
horizontal components of the projectile. We can do this by using the
formula Δdy=K x Δdxn. But we do not have the values of K and n. So we
will be using the kinematic equations to find the missing values.In this
equation Δdy=VyΔt+1/2aΔt2, we know that the initial velocity is zero. So we
can make the equationΔdy=1/2aΔt2. We can use the same equation for the
horizontal component. Δdx=VxΔt+1/2at2. We know that the acceleration is
zero. So we can make the equationΔdx= VxΔt. From this equation, we
know that Δdx is directly proportional to Δt. We can substitute Δd
x for Δt in
Δdy=1/2aΔt2. So we get Δdy=1/2a(VxΔdx)2. After simplifying the equation, we
get Δdy=a/2Vx2(Δdx)2 and our K=a/2Vx2 and our n=2.
If we make a graph between Δdx and Δdy, we get an exponential graph. But
in order to get the slope from the graph we have to find the proportionality
between Δdx and Δdy. We can use the n value (2) to change the Δd
x so that
2
it is proportional to Δdy. The proportionality would beΔdyαΔdx .By doing
this, we have linearized the graph. The K value can be calculated using the
graph.
From what we know, the base graph would have an exponential function
because the vertical drop and horizontal displacement are directly
proportional and the n value is 2. The path of the object will be downwards
with constant horizontal velocity and consistently changing vertical velocity.
METHOD AND APPARATUS
All the apparatus and instructions related to the method are listed on
“TRAJECTORIES 1 DATA ANALYSIS ”,SPH4U1,Sinclair,Fall 2022. The only
deviation in the experiment was not using the plumb bob since it was not
available.
OBSERVATION
The horizontal and vertical displacements of the object are the following;
Number Horizontal
displacement
(Δdx)
Δdy
Δdy
Δdy
Average
Δdy
Trial
1
Trial 2 Trial
3
1
4
0.5
0.4
0.6
0.5
2
8
1.2
1
1.1
1.1
3
12
4.2
3.9
3.7
3.9
4
16
8.5
8.8
8.9
8.7
5
20
14.2
14.7
14.9
14.6
6
24
20.8
20.7
20.9
20.8
7
28
32.3
31.9
32.1
32.1
We can see from the table that as the horizontal displacement increases
the vertical displacement also increases.
Following is the graph between the horizontal and vertical displacement;
-1
We can see that the slope is equal to 0.04cm
.
ANALYSIS
It is clearly visible from the table that as the x-axis(horizontal displacement)
increases, the y-axis(vertical displacement) also increases. That means
that the relationship between is directly proportional. As one variable
increases, the other follows.
MATHEMATICAL ANALYSIS
From the table, we know that the vertical displacement increases along
with horizontal displacement. So, they are directly proportional to each
other.
Δdy α Δdx
If we change the proportionality to equality, we have to use the equation
Y=Kxn,
Δdy =K x Δdxn
We will be using the log formula to calculate the n value,
n= log(y2/y1)/log(x2/x1)
We will take 2 points from the graph to calculate the n value
(x1,y1)=(20,14.6)
(x2,y2)=(24,20.8)
Use the two points in the log formula to calculate n,
n=log(20.8/14.6)/log(24/20)
n=2
Now that we know n, we can solve for K,
K=Δdy/Δdx2
K=32.1/282
K=0.04
Final equation
Δdy=0.04 x Δdx2
LINEARIZATION
The graph had a slope that was not linear. The graph has an exponential
function but we need a linear function. So in order to linearize it, we have to
change the values of the x-axis. From the final equation in the
mathematical analysis, we know thatΔdy = 0.04 x Δdx2. If we change from
equality to proportionality, we get
Δdy α Δdx2
So we squared the x values to obtain the proportionality
. This means that
original x values went from 4,8,12,16,20,24,28 to 16,64,144,256,400,576,
784. This is how we linearized the graph.
Discussion
Question No.1)What is the shape of trajectory of a ball launched
horizontally?
Since the ball was experiencing projectile motion, it had a constant velocity
in the horizontal direction and a constantly changing velocity in the vertical
direction. The trajectory has the shape of a halfway parabola. And this
statement is true for all objects that experience projectile motion which
includes the motion of a ski jumper, a rifle bullet or a diver.
Question No.2)What is the relationship between the vertical drop and
horizontal displacement?
Since we know that the vertical displacement is directly proportional to the
horizontal displacement, the vertical displacement will change in the same
way the horizontal displacement changes. If the horizontal displacement
increases, the vertical displacement also increases. If the horizontal
displacement decreases, the vertical displacement also decreases. Since
we know the equationΔdy =0.04 x ΔDx2, we know that the vertical drop will
be 0.04 of the squared value of horizontal displacement.
Sources of error
It is possible that there might be some measurement error in the
experiment because the steel ball was not giving an exact mark on the
impression tape. It was minorly sliding on the tape which made it hard for
us to get an exact point.
Conclusion
We needed the relation between the components of displacement. So, we
set up a ramp and let a steel ball go through at different horizontal
displacements(4cm,8cm,12cm,16cm,20cm,24cm,28cm) and calculated the
vertical displacement. From the data, We learned that as the horizontal
displacement increased, the vertical displacement also increased. Based
on that, we can conclude that the horizontal displacement and vertical
displacement are directly proportional to each other(
Δdy α Δdx).We do know
that both the components are directly proportional but we do not know what
is the exact relation between them. So, we used the proportionality
equation(Y=Kxn) and logarithmic formula (n= log(y2/y1)/log(x2/x1)) to find the
values of n and then K to form a new equation between the displacement
components. The final equation isΔdy=0.04 x Δdx2. The graph between the
components was not linear so we used yα 𝑥2 to linearize it. Applying this
function made the graph go from an exponential function to a linear
function.