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PHYS 141 Introduction to Physics I

Lab #3: Ballistic Motion

Background

In studying the ballistic motion of an object flying through the air,

Galileo Galilei was one of the first people to hypothesize that its motion

is a combination of uniform horizontal motion (�! = constant) and free

fall vertical motion (�" = �, downward). Under this model the

resulting path of the object is a parabola. In this lab we measure the

trajectory of an object flying through the air and compare this motion to

Galileo’s model.

Measurement

A steel ball is accelerated down a mounted J-shaped tube and exits the

tube with velocity �⃗#. Fix the tube position so that its end is as

horizontal as possible, and the ball exits it at (or just off) the edge of the

table. Use masking tape to mark the point on the floor directly below

the tube opening for reference. To measure the path of the ball after it

exits the tube, a vertical stand with waxed tape affixed is placed in the

ball’s path, first at the point the ball exits the tube, and then moving

back by 10 cm increments to 1 m. For each position, at least 3 trials of

the ball marking the wax paper as it collides with the stand should be

made. Use a pen or pencil to annotate the marks so that you know

which marks correspond to which horizontal distance. It will also be useful to mark the floor

below where it hits at a few distances. After reaching 1m, measure the wax tape marks so that

you have a set of (�, �) coordinates of the ball in flight.

Theory

If we assume that the initial motion of the ball exiting the tube (set as � = 0) was in the �-direction:

�⃗# = �#�̂, and that it experiences a constant downward acceleration: �⃗ = ��̂(note that downward

is taken as the positive �-direction), its trajectory is given by the vector equation:

�⃗(�) = �(�)�̂+ �(�)�̂= �⃗# + �⃗#� + $

%

�⃗�% = �⃗

# + �#��̂ + $

% ��̂�% (1)

We can simplify this by taking �⃗# = �#�̂+ �#�̂= 0 as the point where the ball exits the tube. In

components, its trajectory then reads:

� = �#� � = $

% ��% (2)

Note that the horizontal displacement D� = �(�) = �#� grows uniformly with the time of flight,

while the vertical drop distance, D� = �(�) = 1⁄2��2 increases as the square of time. Eliminating

the time � from these equations yields the parabolic path of the ball as it flies through space:

�(�) = &

%((!)" �% ⇒ 5� = *&/%

(!

� (3)

In Your Notebook

Your lab report

should start with a

sentence or two

summarizing main

goal(s) of a lab.

In Your Lab Report

Use sketches and a

caption to describe

the measurement set- up and how you

measure the data you

record.

In Your Lab Report

Your lab report

should have all your

measured data in a

table.

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PHYS 141: Introduction to Physics I page 2 Lab 3: Ballistic Motion

Data Analysis

Plot � vs. � from your data, being sure to label the plot axes. Check

whether it is consistent with the parabolic theory. Sketch a parabola

fitting to the points and predict from this parabola how far from the end

of the tube the ball would land on the floor.

Sketching or fitting to a parabola is difficult, but sketching or fitting to

a straight line is much easier. Based on equation (3), make a second plot

of 5� vs. � and fit it to a straight line. Use this fit to predict where the

ball would land on the floor. In addition, use this fit and an assumed

free fall acceleration of � = 9.8 m/s2 to find �#.

Prediction and Error Analysis

Based on your data from � = 0 to � = 1m and your analysis, make a

prediction about where the ball will hit the floor. Your marks of the

position directly below the ball’s impacts will help to locate the plane

of its motion. After informing your instructor of the prediction,

measure this directly. Explain what sources of experimental and

theoretical errors could account for any discrepancy between your

prediction and the final measurement. For each experimental error

estimate the maximum number of cm this error could reasonably account for in the discrepancy

between prediction and result. Do your errors account for the measured discrepancy? Are there

ways of reducing these errors?

Figure from Galileo Galilei’s Dialogues Concerning Two New Sciences, 1638

In Your Lab Report

Present a plot of both

� vs. � with parabolic

fit and '� vs. � with

linear fit. Calculate

�! based on your

linear fit.

In Your Lab Report

Explain the basis for

your prediction and

discuss any

experimental and

theoretical errors.

In Your Lab Report

Make a claim about

where the ball should

hit the floor.