John McCarthyPage 1

Measuring the Acceleration due to gravity using a projectile launcher

John McCarthy

Aim:

To measure the acceleration due to gravity by calculating the acceleration of a sphere on its upward flight using the equations of motion.

Method:

Equipment:

  • 1 x Aluminium Sphere
  • 1 x Projectile Launcher
  • 1 x 1 metre Ruler
  • 1 x Light Gate
  • 1 x Data Logger
  1. Attach the projectile launcher to a bench and attach the light gate, connected to the data logger, to the projectile launcher.
  2. Line up the base of the 1 metre ruler with the light gate so that measurement of the height that the sphere reaches begins as the sphere passes through the light gate.
  3. At eye level with the ruler, to minimize experimental error, launch the aluminium sphere from the projectile launcher 10 times, taking care to record the height that was reached, and the time that the sphere took to pass through the data logger.
  4. Measure the diameter of the aluminium sphere.

Data:

Diameter of aluminium sphere: 0.018 metres.

Measurements of the height and ball diameter are taken with an accuracy of 3 decimal places, and the measurement of the time taken for the sphere to pass through the light gate is taken with an accuracy of 5 decimal places.

Light Gate Time (s) / Height Reached (m)
0.00773 / 0.245
0.00721 / 0.290
0.00727 / 0.285
0.00764 / 0.260
0.00788 / 0.250
0.00730 / 0.270
0.00830 / 0.220
0.00739 / 0.265
0.00761 / 0.290
0.00723 / 0.300

Data Analysis:

To find the acceleration due to gravity, you must find the initial speed of the sphere as it leaves the projectile launcher. To calculate the initial speed, the formula:

Is used, for example:

From this, we can calculate the initial speed of all 10 launches:

Light Gate Time (s) / Height Reached (m) / Initial Speed (ms-1)
0.00773 / 0.245 / 2.329
0.00721 / 0.290 / 2.497
0.00727 / 0.285 / 2.476
0.00764 / 0.260 / 2.356
0.00788 / 0.250 / 2.284
0.00730 / 0.270 / 2.466
0.00830 / 0.220 / 2.169
0.00739 / 0.265 / 2.436
0.00761 / 0.290 / 2.365
0.00723 / 0.300 / 2.490

To find the acceleration of the sphere as it flies through the air you must find the average value of the height and initial speed.
To calculate the average height and initial speed, the following formulae are used:

And hence:

Using these values, along with the knowledge that at the top of its flight, the spherical ball with have an instantaneous velocity of 0ms-1, we can find a measure of the acceleration due to gravity. As we do not have a value for the time taken, we need to use the equation of motion:

And if we substitute V, the final velocity, as 0, u, the initial velocity, as our average 2.3868, and s as our average height reached 0.2675, we attain the equation

That is, 10.65ms-2 downwards.

Evaluation:

The final value attained for the acceleration due to gravity was 10.65ms-2downwards, however the actual value for the acceleration due to gravity is 9.8ms-2 downwards.

There are a number of experimental errors that could contribute to this inaccuracy.

Some systematic errors, errors that are constant throughout the experiment, include:

  • Friction from the launcher slowing the ball down as it is released from the projectile launcher.
  • Air resistance affecting the flight of the sphere as it rose through the air.
  • An incorrect measurement of the diameter of the ball.
  • An incorrect measurement of where the base of the ruler is in relation to the light gate.
  • An incorrect configuration of the data logger giving faulty results for the time taken for the sphere to pass through the light gate.
  • An incorrect or faulty light gate.

Some random errors, errors that are not constant throughout the experiment, that could have occurred include:

  • Wind or air pressure from external sources affecting the flight of the sphere unevenly throughout the sequence of tests.
  • Inaccurate judgement of the maximum height that the sphere reached during its flight.

Whilst the experiment was capable of measuring the acceleration of the sphere with reasonable accuracy, there are a number of possible improvements that can be made to the experiment to attain more accurate results. These include:

  • Filming the flight of the sphere with a slow motion camera, and then analysing the flight of the sphere in each through with video capture software after the experiment so that a more accurate value for the height that the sphere reached can be attained.
  • Running the experiment in a partial or total vacuum.
  • Using an electronic measurement system to find the diameter of the sphere to avoid human error.