How does the height of a drop affect the time it takes an object to hit the ground?

• Higher drops take longer.
• Physics gives us an equation to predict the relationship.
• Statistics lets us test the equation with data.

• Objects accelerate toward Earth at about 9.81 m/s² near the surface.
• If we ignore air resistance, a dropped object starts from rest and speeds up as it falls.
• Falling time does not increase at a constant rate with height.

Prediction: A plot of height versus time squared should be a straight line.

For an object released from rest with constant gravitational acceleration:

\[ h = \frac{1}{2}gt^2 \]

where:

• \(h\) = drop height (meters)
• \(g\) = gravitational acceleration (m/s²)
• \(t\) = falling time (seconds)

Solving for time gives \(t = \sqrt{2h/g}\).

A simple linear regression has the form:

\[ y = \beta_0 + \beta_1 x + \varepsilon \]

For our falling-object data, set \(y=h\) and \(x=t^2\):

\[ h = \beta_0 + \beta_1t^2 + \varepsilon \qquad\text{with}\qquad \beta_1 \approx \frac{g}{2} \]

Thus, estimated gravity = 2 × regression slope. The intercept should be near zero.

This presentation uses 10 simulated drops, one from each height between 1 and 10 meters. Each falling time has a small simulated measurement error. These are not actual experimental measurements.

Here is the R code that fits a straight line to height versus time squared:

# drops contains height, time, and time_squared
model <- lm(height ~ time_squared, data = drops)
summary(model)

# Since slope = g/2, estimate gravitational acceleration:
g_estimate <- 2 * coef(model)["time_squared"]

The lm() function estimates the slope and intercept.

The points lie close to the fitted straight line, as the free-fall equation predicts.

A residual is the difference between an observed height and the height predicted by the regression.

Residuals near zero mean the predictions are close to the simulated measurements.

Hover over a point to see the simulated measurement. You can also zoom and pan.

• The fitted regression slope was 4.870 m/s².
• The intercept was 0.057 m (ideally 0).
• The estimated gravitational acceleration was 9.740 m/s².
• The regression had R² = 0.9987, so the linear model explained 99.9% of the variation in height.
• The gravity estimate differed from 9.81 m/s² by 0.72%.

The small simulated timing errors make the estimates slightly different from the ideal values.

• Height and time squared have an approximately linear relationship, consistent with the free-fall equation.
• Simple linear regression provides a way to estimate gravitational acceleration from falling-time data.
• Residual plots help us check how well a straight line fits the observations.
• The analysis illustrates the physics principle, but simulated data do not independently verify gravity. A real experiment would require measured drop times.