Data-based question 1: Investigating Projectile Motion

A student investigates the motion of a small steel ball projected horizontally from a table. The height (H) of the table is 1.25 m. The horizontal distance (R) travelled by the ball before hitting the ground is measured for different initial horizontal speeds (u). Air resistance is considered negligible.

The data collected are shown in the table below. Uncertainties in measurements are not shown.

(a) (i) State the equation linking horizontal distance (R), initial horizontal speed (u), and time of flight (t). [1]
(ii) State the equation linking vertical height (H), acceleration due to gravity (g), and time of flight (t), assuming the initial vertical velocity is zero. [1]

(b) Using the data provided and the equations from (a), determine the time of flight (t) for the steel ball. Assume (g = 9.8 text{ m s⁻}²). [2]

(c) The student recorded data for (R) and (u).
(i) Interpret the provided diagram of the expected graph of (R) against (u) on the axes below. Briefly describe the general trend indicated by the diagram. [1]
[VISUALIZATION: Graph axes with (R) on the vertical axis and (u) on the horizontal axis. Axes start from zero. No data points plotted. Enhanced description: Expect a straight line indicating a constant rate of increase, which illustrates the direct proportionality between (R) and (u).]
(ii) Explain the relationship between (R) and (u) as indicated by the diagram. [2]

(d) The student repeats the experiment but launches the ball from a height of 2.50 m (double the original height) with an initial horizontal speed of 4.0 m s⁻¹. Estimate the new horizontal distance travelled by the ball. [3]

(e) Calculate the gradient of the (R) against (u) graph using the provided data. [1]

(f) State the physical meaning of the gradient calculated in (e). [1]