physical lab

Experiment 3

PROJECTILE MOTION

Objective

To use kinematics equations to predict the range and time of flight of the iOLab sensor when

undergoing projectile motion, and to verify these equations by comparing the predicted values to

the measured range and time of fight.

Theory

Consider a cart undergoing projectile motion, as shown in Figure 1. The kinematics equation for

the horizontal position of the cart is:

𝑥 = 𝑥𝑜 + 𝑣𝑜𝑥𝑡 + 1

2 𝑎𝑥𝑡

2 (1)

We can assume that the effect of air resistance on the cart is negligible; hence, no horizontal forces

act on the cart while it is in flight, so the cart has no horizontal acceleration (ax = 0) and its

horizontal velocity stays constant (v0x = v).

If the horizontal displacement of the cart is measured from the edge of the launch to the point of

impact (cushion – see Figure 1), we can set x0 = 0 and let x = Rth, the (theoretically determined)

range. Equation (1) then becomes:

𝑅𝑡ℎ = 𝑣𝑡𝑡ℎ (2)

where tth is the theoretical time of flight of the cart.

Figure 1 – The flight path of a cart undergoing projectile motion.

Cushion

v, initial velocity

R, range

h, height

Table edge

y

x Floor

The time t is calculated in terms of the cart’s vertical displacement from the edge of the table to its

point of impact on the cushion. The kinematics equation for the cart’s vertical position is:

𝑦 = 𝑦𝑜 + 𝑣𝑜𝑦𝑡 + 1

2 𝑎𝑦𝑡

2 (3)

The vertical component of the cart’s motion is identical to the motion it would have if it were

dropped from rest from the same height, h, above the cushion. A cart dropped from rest has an

initial velocity v0y = 0 and an acceleration ay = g, where g is earth’s gravity (9.81m/s 2).

If we take the table edge to be y0 = 0 and substitute these values into Equation (3), then solving for

the (theoretically determined) time of flight tth gives:

𝑡𝑡ℎ = √ 2ℎ

𝑔 (4)

Where h is the vertical distance the cart travels, as shown in Figure 1.

Equations (2) and (4) are used to predict the range and time of flight of the cart when it is a

projectile.

Apparatus

iOLab sensor

iOLab data acquisition program

access to a table or desk to project the iOLab off of

pillow or other way to cushion the iOLab upon impact

tape measure or other way of measuring distances of 1 to 2 metres

Procedure

Before you start

The iOLab should be set wheels-down on the table so it rolls like a cart. While the Wheel > Velocity and Accelerometer data is being captured, you will roll the iOLab

sensor off the edge of the table onto the cushion.

Do a test run to ensure the cart will land safely on the cushion.

The height that the cart falls will be measured experimentally, as will the horizontal range that the cart travels through.

The iOLab’s horizontal velocity v, just as it leaves the table edge, can be measured with the data acquisition program. The time the cart spends in projectile motion can also be measured.

A typical graph of such motion might look like this:

Figure 2 – Typical acceleration and velocity data for an iOLab device undergoing projectile motion.

As indicated in Figure 2, the launch velocity of the sensor v can be obtained by hovering the cursor over the point in the velocity-time graph where the sensor left the table.

The experimental time of flight for the sensor can likewise be measured by selecting the period of time that the sensor was in the air, as determined by when the accelerometer registers an

acceleration of 0 m/s2 (where the cart is in free-fall).

Data Collection

Take all necessary measurements to determine the predicted theoretical time-of-flight tth, the predicted theoretical range Rth, the experimental time-of-flight texp, and the experimental range

Rexp. Be prepared to do several runs if you don’t get good results the first time.