CP1a – Acceleration of a Freely-Falling Object

Theory & Background

Understanding the physics behind free fall and the SUVAT equations

Objectives

Measure the acceleration due to gravity g of a freely-falling object using a trapdoor and electromagnet
Plot a graph of t² against h and determine the gradient to calculate g
Evaluate experimental uncertainty and calculate percentage difference from the accepted value

The Physics

When an object is released from rest and falls freely under gravity (ignoring air resistance), it accelerates uniformly. This means we can apply the SUVAT equations of motion.

For our experiment, the object starts from rest (u = 0), falls a height h, and takes time t. The relevant SUVAT equation is:

s = ut + ½at² Since u = 0, s = h, and a = g : h = ½gt² ∴ t² = (2/g) · h

Comparing with y = mx + c → plotting t² (y-axis) against h (x-axis) gives a straight line through the origin with gradient m = 2/g, so g = 2/m

This linear relationship is the key to the experiment. By measuring multiple (h, t) pairs and plotting t² vs h, the gradient directly gives us g.

Method A — Trapdoor & Electromagnet

A steel sphere is held by an electromagnet. When the circuit is broken, the timer starts simultaneously. The sphere falls and strikes a trapdoor below, breaking a second circuit and stopping the timer. The time recorded is the time of free fall over height h.

Safety Considerations

⚠️ Ensure all apparatus is secured and cannot topple over during the experiment.

🪨 Be aware of falling objects — do not place hands under the drop path.

Step-by-Step Procedure

Follow these steps in the Simulation tab to collect your data

Equipment (Method A)

Steel sphere (5–10 mm dia.) · Electromagnet · Electronic timer · Trapdoor · Metre rule (mm resolution)

Set the drop height

Use the slider in the Simulation tab to set the height h from the bottom of the sphere to the trapdoor. Start at around 0.20 m.

Drop the sphere & record time

Click Drop Sphere. The electromagnet releases the sphere and the timer starts. Record the time t shown when the sphere hits the trapdoor.

Repeat 3 times at the same height

Each drop includes a small random error simulating real measurement noise. Record all three times and the simulation will calculate the mean automatically.

Record the reading

Click Record Reading to add the mean time for this height to your data table. You need at least 6 different heights.

Change the height and repeat

Adjust the height slider and repeat steps 2–4 for at least 6 different heights between 0.20 m and 1.00 m.

Analyse your results

Go to the Data Table tab to view your recorded values. Then visit Graph & Analysis to plot t² vs h and determine g.

g = 2 ÷ gradient of t² vs h graph

Controls

Drop height 0.50 m

        0.20 m1.00 m
      

Drop 1

— ms

Drop 2

— ms

Drop 3

— ms

Mean t

—

Drops at this height: 0/3

Readings recorded: 0/6 minimum

Results Data Table

Auto-filled from the Simulation tab. You can also edit cells directly or add rows manually. Maximum 6 readings.

🔬

Step 1 — Simulate Go to Simulation tab, set height, perform 3 drops, click Record Reading

→

📊

Step 2 — Check Data Switch to Data Table to verify your reading. Edit any cell if needed.

→

🔁

Step 3 — Repeat Return to Simulation, change the height, and repeat until you have 6 readings.

#	h/ m	t₁/ ms	t₂/ ms	t₃/ ms	t̄/ ms	t̄/ s	t̄²/ s²	Range/ ms	½ RangeΔt / ms	%U in t
No data yet — go to the Simulation tab and collect readings, or add a row manually below.

Graph & Analysis

Plot of t² (y-axis) against h (x-axis). The gradient m gives g = 2/m

Calculated g

— Collect ≥6 readings first

Graph Parameters

Gradient m —

R² (fit quality) —

Accepted g 9.81 m s⁻²

% Difference —

Uncertainty Estimate

Avg %U in t —

%U in t² —

%U in g —

Note: %U in t² = 2 × %U in t (since t is squared)

Interpretation

Complete your experiment to see the analysis here.

Discussion Questions

Write your answers below. Click "Show Model Answer" when you're ready to compare.

Question 1

Explain why the graph of t² against h should be a straight line passing through the origin.

From SUVAT: h = ½gt², rearranging gives t² = (2/g)h. This is of the form y = mx where m = 2/g is constant. Since g is constant throughout the fall, the relationship between t² and h is directly proportional — a straight line with a constant gradient passing through the origin. Any deviation from a straight line would indicate non-uniform acceleration.

Question 2

Discuss the effect of air resistance on your measured value of g. Would your result be greater than, less than, or equal to 9.81 m s⁻²? Explain your reasoning.

Air resistance acts upward, opposing the motion of the falling sphere. This means the net downward force is less than mg, so the actual acceleration is less than g. As a result, the sphere takes longer to fall the same height — the time t is larger than it would be in a perfect vacuum. Since t is larger, t² is larger, the gradient m = 2/g is larger, and therefore the calculated g = 2/m will be smaller than 9.81 m s⁻². The measured value will be an underestimate.

Question 3

Give two sources of uncertainty in this experiment and suggest how each could be reduced.

1. Timing uncertainty: The electronic timer may not start/stop precisely when the sphere is released/hits the trapdoor due to mechanical delays. This can be reduced by using a more sensitive electromagnetic trigger or a light gate, and by repeating measurements and taking a mean. 2. Height measurement uncertainty: The height h is measured with a metre rule to ±1 mm, but the reference point (bottom of sphere) may be difficult to define precisely. This can be reduced by using a travelling microscope or taking multiple careful measurements of h.

Question 4

The percentage uncertainty (%U) in t² is twice that in t. Explain why, using the propagation of uncertainty rules.

When a quantity is raised to a power n, the percentage uncertainty is multiplied by n. Since t² involves squaring t (power of 2), the percentage uncertainty in t² = 2 × %U in t. This comes from the general rule: if y = xⁿ, then %U(y) = n × %U(x). So if the uncertainty in t is 1%, the uncertainty in t² is 2%.

Question 5

Your calculated value of g has a percentage difference from 9.81 m s⁻². Comment on the accuracy and precision of your method based on your results.

Accuracy refers to how close the result is to the true value (9.81 m s⁻²). The percentage difference gives a measure of accuracy — a small %D indicates good accuracy. Precision refers to the spread of repeated readings — a small %U in t (from the half-range of repeat drops) indicates good precision. A result can be precise but inaccurate (systematic error) or accurate but imprecise (large random errors). To improve accuracy: eliminate air resistance effects by using a denser sphere; to improve precision: use more repeats and a more sensitive timer.