Experiment 1
PROJECTILE MOTION
OBJECTIVE
The goal of this lab is to get the student (you) familiar with the relationship between position, velocity, and acceleration.
APPARATUS
Due to the online nature of this lab, the “apparatus” principally used is an online resource produced by PhET Interactive Simulations that can be found at PhET.
BACKGROUND Velocity is the change in an object’s position divided by the time interval spanned to yield that change. In the horizontal direction—the x direction—the average velocity1 can be written as vx = ∆x ∆t, (1) and in the vertical direction—the y direction—the average velocity can be written as vy = ∆y ∆t. (2) Notice that equations (1) and (2) are actually the same mathematical equation, but different symbols are used for convenience. In our case, the symbol x is used to denote the horizontal direction, and the symbol y is used to denote the vertical direction. But velocity is velocity, whether in the horizontal or vertical direction. Velocity is merely a definition. If you’ve taken algebra, you’ll recognize the form of (1) and (2): the change in one quantity divided the by change in another quantity. What is this quotient typically called in a mathematics class? Slope. Therefore, we see that: Velocity is the slope of a position-versus-time graph. velocity = change in position change in time = slope Refer to Figure 1 for an illustration of this. Notice that a slope can be positive or negative. The sign of the velocity indicates the direction of the velocity. For example, consider that you are watching a car on the road. When it is going away from you, its velocity is positive. When it is coming toward you, its velocity is negative.2 Think about the reason why. 1The instantaneous velocity in the x-direction can be found by taking the limit: vx = lim∆t→0 ∆x ∆t = lim∆t→0 x (t + ∆t) − x(t) ∆t = dx dt. Similarly, for the y-direction and any other direction. 2This, of course, depends on how you define your coordinate frame. But assuming you stand at the origin, the convention mentioned applies.
Figure 1: A vertical position vs. time plot. The orange segments represent the slope—the velocity—at the points. Each point is used as an example. What about acceleration? Acceleration is also defined as a change in something—a change in velocity. In the x direction the average acceleration can be written as ax = ∆vx ∆t, (3) and in the y direction the average acceleration can be written as ay = ∆vy ∆t. (4) Therefore, we see that: Acceleration is the slope of a velocity-versus-time graph. accleration = change in velocity change in time = slope PROJECTILE MOTION In the absence of air resistance3, all objects dropped near the earth’s surface accelerate toward the earth with the same constant acceleration under the influence of the earth’s gravity. This specific gravitational acceleration, since used so frequently for near-earth predictions, is denoted with its own symbol g and has an approximate magnitude of g = 9.8 m/s. (5) Without gravity, a rock could be tossed skyward at an angle and it would follow a straight-line path. With gravity, however, the path curves. If an object—a soccer ball, for example—is kicked at some 3 In many problems, neglecting air resistance is a good approximation; in others, however, this approximation falls short. In future physics courses, air resistance will be accounted for, or at least there will be some sort of quantification as to whether it is significant for the problem at hand, but in this lab—and for most of this course, unless stated otherwise—air resistance will be assumed to be negligible (especially for cases when more than conceptual understanding is required).
Figure 2: An object undergoing projectile motion. The length of the arrows denotes the magnitude of the vector in that direction. angle relative to the ground, we say that the object has a component, or piece, of its velocity in the x direction (horizontal) and a component of its velocity in the y direction (vertical), which can be denoted as vxi and vyi, respectively. This is shown in Figure 2. As time progresses, the object moves forward and upward until it reaches its zenith, and then it continues to move forward and downward. The flip in vertical direction is caused by the downward tug of gravity, which accelerates the object toward the ground. Notice that the horizontal velocity remains constant throughout the motion. In other words, the initial velocity in the x direction is equal to the final velocity in the x direction. In equation form, vxi = vxf. This is because gravity acts only in the vertical direction, meaning that the horizontal direction experiences no acceleration. This type of simple motion is known as projectile motion, which is motion experienced by an object in free fall. The equations of motion describing our specific scenario are as follows: Horizontal motion: xf = xi + vxi t, (6) and Vertical motion: yf = yi + vyi t − 1 2
PART 1: Finding the Slope of Position
1. Open the PhET Interactive Simulation for projectile motion. A menu with four options will appear. Select the final option entitled “Lab.” An interface will appear. Leave all of the variables at their default values.
2. Click on to launch a projectile. A blue parabolic curve will be traced out by the software, with points marked out at specific time intervals.
3. Open an Excel spreadsheet (or a MATLAB script, Mathematica notebook, or any preferred plotting software). If using Excel, create three columns: one for time, range, and height.
4. Select the target scope, which is found in the upper right of the general PhET interface. Use this target scope to find the time, range (horizontal position), and height (vertical position) of every point along the curve by hovering the crosshair over each point. Record these values in the Excel spreadsheet you created. Do not record the single point where the object hits the ground. An example of what your table will look like is shown below. Time Range Height . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Create two scatter plots: (i) Horizontal position (range) versus time. (ii) Vertical position (height) versus time. Your plots should look similar to those seen in Figure 3. You must turn in your plots with the rest of this manual.
6. Based on the horizontal position-vs-time plot, is the horizontal speed constant throughout the object’s motion? Explain how the plot supports your answer.
7. Based on the vertical position-vs-time plot, is the vertical speed constant throughout the object’s motion? Explain how the plot supports your answer.
8. Pick two points on the horizontal position-vs-time plot. Using equation (1), calculate the velocity in the x direction. Show your work. vx = m/s. (i) Does the value you calculated represent the velocity through the entire trajectory? Explain your answer.
9. Pick two points on the vertical position-vs-time plot. Using equation (2), calculate the velocity in the y direction. Show your work. vy = m/s.
(i) Does the value you calculated represent the velocity through the entire trajectory? Explain your answer.
PART 2: Finding the Slope of Velocity
1. Using the data, you already collected and tabulated in the previous part, apply equations (1) and (2) to find the approximate velocity at points in time. It might be helpful to create a second table with three columns: time, velocity in the x direction, and velocity in the y direction. And example of what your table will look like is shown below. Time Horizontal Velocity Vertical Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Using the new table, you created, generate two scatter plots. The new plots should appear similar to Figure (4). You must turn in your plots with the rest of this manual. Figure 4: Plots of velocity versus time.
3. Pick two points on the horizontal velocity-vs-time plot. Using equation (3), calculate the acceleration in the x direction. Show your work. ax = m/s2.
(i) Does the value you calculated represent the velocity through the entire trajectory? Explain your answer.
(ii) Based on your data, is the acceleration roughly constant? Explain how you know.
4. Pick two points on the vertical velocity-vs-time plot. Using equation (4), calculate the acceleration in the y direction. Show your work. ay = m/s2.
(i) Does the value you calculated represent the velocity through the entire trajectory? Explain your answer.
(ii) Is the value negative? If so, how can you interpret that? If not, should it be? (iii) Is the acceleration roughly constant? How can you tell?
PART 3:
Questions 1.
A tennis ball is hit at some angle relative to the ground, giving it a vertical velocity of 10 m/s upward and a horizontal velocity of 15 m/s rightward. (Consider upward to positive along the y-axis and rightward to be positive along the x-axis.)
- Assuming the ground is level, how long will the ball remain in the air? t = s.
(ii) How far will the ball travel horizontally in that time? ∆x = m. 2. Tom looks from his 45-m high balcony to a swimming pool below—not exactly below, but rather 15 m from the bottom of his building. If he wants to jump directly from his balcony to the pool, what is the minimum horizontal velocity he would need to make it safely to the water? vxi = m/s.