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.