Linear Acceleration and Projectile Path

Objective

To study the motion of a body when projected at an angle above the horizontal on an

inclined plane.

Introduction

In this experiment we’ll use an air table to study projectile motion. The air table allows

one or two stainless steel pucks to glide on a glass surface using a cushion of air, similar to

an air-hockey game. If we place a piece of paper underneath the puck, we can record its

trajectory by use of a spark apparatus, which leaves a trail of dots on the paper. More on

this later.

Theory

You may recall the formula for an object’s motion with constant acceleration in one

dimension (x):

x = x◦ + v◦t +1

2at2,

where x is the object’s position at any time t (instantaneous position), x◦ is the displacement

at time t = 0 (initial displacement), v◦ is the speed at time t = 0 (initial speed) and a is

the object’s acceleration. The mathematics for considering motion in a second independent

dimension y follows exactly, but we need to be careful - we have separate accelerations,

displacements and velocities in both x and y, as shown in the general equations below:

x = x◦ + vx◦t +1

2axt

2, (1)

y = y◦ + vy◦t +1

2ayt

2 (2)

Notice how we’ve distinguished between the initial positions, displacements and acceler-

ations of x and y components by the subscripts. These are important, so be sure to keep

track of them so that you know in which direction you’re going!

2

α

v0

vx0

vy0

x0

y0

x

y

FIG. 1: Top and side view of the air table used in the experiment.

How do x and y relate to each other? The diagram in figure 1 should help clear things

up. The quantity v◦ is the total initial speed, which is related to vx and vy through the

launch angle α, by

vx◦ = v◦ cos α and vy◦ = v◦ sin α (3)

To simplify things, we’re going to assume that x◦ = y◦ = 0 for this experiment. This is

ok, because we’re always allowed to pick the origin of our co-ordinate system (once). The x

and y directions will be given by the plane of our air table.

Normally, the air table sits flat and level. You might well realize that in this orientation,

there shouldn’t be any acceleration in either the x or y directions. However, if we tilt the air

table (say along the y axis), we can produce a gravitational acceleration in the y direction

only. This means that ax = 0, and ay (which depends on the tilt of the table) can be written

as ay = −a, where we introduce the -’ve sign to indicate that the acceleration is in the -’ve

y direction.

We can summarize the above by rewriting equations (1) and (2) as

x = (v◦ cos α) t (4)

y = (v◦ sin α) t− 1

2at2 (5)

3

In this experiment you’ll measure the range (R) and maximum y displacement (ymax) of

your puck to see how well the equations of motion above are able to predict the trajectory.

The range is the horizontal distance travelled after y has returned to zero, and is given by

R =v◦

2 sin 2α

a, (6)

and the maximum y displacement is given by

ymax =v◦

2 sin2 α

2a. (7)

You can look at figure 2 for a schematic representation of these quantities.

Apparatus

We make use of a standard air table complete with a puck, a clear plastic launching

ramp and a spark timer for indicating equal time increments dt while the puck is in motion.

The spark timer fires at a fixed rate, leaving small burn marks on the underside of the

paper placed between the puck and glass table. The spacing of these dots is thus directly

proportional to the puck’s speed.

Procedure

Prepare the air table with a clean sheet of paper, checking first that the carbon paper is

free of scratches or tears. Make sure that the paper is flush with the bottom of the table

and that it is smooth.

There are actually two pucks on the air table. The extra puck should be immobilized on

the table, using the coral provided, and its tubing kept out of the way. It’s very important

that you keep this second puck on the air table, as the spark timer won’t function without

it.

Position the “launching ramp” in a bottom corner of the table with the side marked AG

horizontal so that the side marked CD is inclined to the horizontal, as in figure 2. Mark

the two lines CD and DE so that the lines can be transferred to the other side of the paper.

The value of the launch angle α is approximately 50◦, but you should measure it yourself to

be certain. Place the piece of sponge rubber along the bottom edge of the table opposite

4

R

ymax

α

A G

B C

D E

F

FIG. 2:

to the launching ramp, to absorb the shock of the landing puck. With the air on, make

one or two trial launches of the test puck from the ramp CD, to get a feel for the system.

It’s important to release the puck BEFORE it has left the ramp, to ensure that you’re not

interfering with it during its “flight”.

Turn on the spark timer and select a sparking interval (dt) of 50 milliseconds, and make

a note of it in your log. The timing of the sparks is accurate to ±1%, which corresponds

to ±0.5ms on the 50ms setting. MAKE SURE THAT YOU ARE HOLDING THE PUCK

BY ITS VINYL TUBE and you are NOT TOUCHING THE TABLE OR THE OTHER

PUCK OR YOU WILL GET A SHOCK! Hold down the two buttons that start the spark

and launch the puck as you have done in the trials. NOTE: Each student should make

her/his own set of tracings and calculations. Otherwise the workload ends up being uneven,

and you won’t learn as much.

Turn the sheet of paper over and transfer the line CD by tracing through the paper.

Draw a line parallel to CD along the line of spark points which were recorded while the puck

was still in contact with the ramp and mark the position of the centre of the puck just as

the puck left the ramp. This is the point (0,0). Carefully draw through (0,0) parallel to DE.

5

Now you can make the measurements that are requested in the worksheet (Tables 1-3) and

use the recorded values to calculate the requested quantities (Table 4).

6

Name: Partners Name

Student ID Date

WORKSHEET: To be handed in:

1. Constants:

VALUE UNCERTAINTY

Radius of puck (This will effect your choice of (0,0))

Time between sparks

Launch angle, α

Length of air table between feet (hypotenuse)

Height of block used to raise table (opposite)

2. The number of spark timer points until the puck:

NUMBER OF SPARKS FROM ORIGIN UNCERTAINTY

Is back to y = 0 (NR)

Reaches ymax (Nmax)

Reaches the final point (Nfinal)

NOTE: these numbers will not, in general, be whole numbers. You will have to estimate

fractions of intervals, with appropriate uncertainties.

7

3. Displacement measurements:

DISPLACEMENT UNCERTAINTY

Average s in one interval at the instant the puck leaves the ramp

The horizontal range, R

ymax

The (x, y) co-ordinates selected final point

NOTE: s is the distance travelled by the puck in that time interval.

4. Analysis Table:

VALUE UNCERTAINTY

v◦ = s/dt

v◦ = R/(cosα tR)

a (using R and equation 6)

a (using ymax and equation 7)

xf (using equation 4, and average a)

yf (using equation 5, and average a)

sin θ, inclination of air table = opposite/hypotenuse

Be sure to record the systematic error for every measurement you take!

For this experiment it is the systematic errors which determine the final uncertainty

in your results. Consult the notes on error analysis if you’re confused as to how to

propagate your uncertainties.

5. Do the two values for v◦ agree within experimental error? Which is more precise?

6. Compare xf , yf to their measured values.

7. Determine the theoritical value of the acceleration, a and its uncertainty, using a =

g sin θ, where θ is the inclination of the air table.

8. Discuss the agreement of the different determinations of v◦, of a and of xf and yf ,

taking the experimental errors into account. Can you find any evidence of friction in

your results?