Final answer:
Acceleration due to gravity is twice the slope of the distance-time squared graph because this slope represents half of the constant acceleration according to the kinematic equation s = (1/2)at². This is supported by calculus as acceleration is the second derivative of position with respect to time, and the velocity-time graph demonstrates the constant acceleration by its straight-line slope.
Step-by-step explanation:
The acceleration due to gravity is equal to twice the slope of the distance-time squared graph because of the kinematic equation for uniformly accelerated motion, specifically when acceleration is constant. When plotting distance (s) against time squared (t²), the graph's slope equals half of the acceleration (a) due to the relationship s = (1/2)at². Consequently, the acceleration due to gravity (g) can be extracted from such a graph since it is constant near the Earth's surface. This is directly related to acceleration being the second derivative of position with respect to time in calculus.
As further evidence, a velocity versus time graph with constant acceleration shows a straight line indicating the constant rate of change of velocity. Thus, the gradient or slope of the velocity-time graph represents the constant acceleration. On this graph, the rise is the change in velocity (Δv) and the run is the change in time (Δt). Therefore, the acceleration which represents gravitational acceleration when an object is in freefall can be determined by analyzing the slope of these lines.