For an object moving along a straight line, the slope (gradient) of its distance time graph at any point represents which physical quantity?

Introduction / Context:

Distance time graphs are a key tool in kinematics for visualising how far an object has travelled as time passes. The slope or gradient of such a graph carries important physical meaning. This question checks whether you know which physical quantity is represented by the slope of a distance time graph when an object is moving along a straight line.

Given Data / Assumptions:

• The vertical axis represents distance travelled from a chosen origin.
• The horizontal axis represents time.
• The motion is along a straight line, so distance and displacement coincide in magnitude.
• The slope is defined as change in distance divided by change in time (Δdistance / Δtime).

Concept / Approach:

Speed is defined as the rate of change of distance with respect to time, that is speed = distance / time for uniform motion or speed = Δs / Δt more generally. In a distance time graph, the slope at any point is given by rise over run, which is Δdistance / Δtime. This matches the definition of speed. Acceleration would be associated with the slope of a velocity time graph, not a distance time graph. Other quantities such as momentum, mass and force do not directly correspond to this slope in basic kinematics.

Step-by-Step Solution:

Verification / Alternative check:

Consider an object moving at a constant speed of 5 m s^-1. In 1 second it travels 5 m, in 2 seconds 10 m, in 3 seconds 15 m, and so on. Plotting these points gives a straight line whose slope is rise over run = 5 / 1 = 5, which numerically matches the speed. If the object speeds up, the graph becomes steeper, and if it slows down, the graph becomes less steep, which is exactly how you would expect speed to behave on the graph. This confirms the identification of slope with speed.

Why Other Options Are Wrong:

Option A (acceleration): Acceleration is the rate of change of velocity with time and is represented by the slope of a velocity time graph, not a distance time graph.

Option B (momentum): Momentum depends on mass and velocity, not directly on distance versus time slope in this simple graph.

Option C (mass): Mass is a property of the object and does not appear as a slope in a distance time graph.

Option E (force): Force is related to acceleration via F = m * a and again does not directly correspond to the slope of distance versus time.

Common Pitfalls:

Students often mix up the meanings of slopes on different types of graphs. A simple rule to remember is: on a distance time graph, slope is speed; on a velocity time graph, slope is acceleration; and the area under a velocity time graph gives displacement. Keeping these associations clear will help avoid mistakes in both conceptual and numerical questions involving graphs.

Final Answer:

The slope of a distance time graph represents the speed of the object.