# What can you determine from the calculation of the slope of a distance-time graph?

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In a space-time graph of a movement, the slope represents the speed of the movement.

The equation of a straight line has the form y = mx + b, where m is the slope of the line and b is the value of y when x = 0.

For a uniform rectilinear motion, the graph is a straight line and the distance equation for this case has the form d = vt + d0, where v is the speed and d0 is the initial position.

Assuming that the distances are taken on the Y axis and time on the X axis, by comparing the two equations, we see that the value of the speed matches the value of m, which is the slope of the graph. The value of b coincides with the initial position, ie, the value of d when t = 0.

y = mx + b

d = v t + d0

Then: v = m, d0 = b

This same analysis is valid if the motion is varied, where the graph is a curved line; in this case the curve, representing the movement, has a straight line tangent at each point and the slope of this straight line represents the instantaneous speed of the movement at that point.

The slope of the distance-time graph refers to the speed of an object.

If the graph is steep, then, it has a high slope. This means that that the object is moving at high speed.

Also, if the graph is a horizontal line, its slope is zero. This means that the object has stopped or stationary.

Moreover, the signs of the slope of the distance-graph refer to the direction of the object.

If the slope of the graph is positive, it means that the object is moving away from its starting point.

And if the slope is negative, it indicates that the object is returning to its starting starting point.

For a distance-time graph, the distance is on the y-axis and time is on the x-axis. Consider what your units are when you are finding the slope. Slope is often taught in school as "rise over run". In other words, you are dividing some amount of the variable on the y-axis by some amount of the variable on the x-axis. Therefore, the units of your slope will be distance units over time units. Let's just use meters for distance and seconds for time. In that case, the units for slope are m/s. What does this remind you of? Probably velocity or speed. To determine which of the two - velocity or speed - it is, remember that you were using *distance*. By definition, velocity is *displacement* over time whereas speed is *distance* over time. Therefore, the slope of a distance-time graph gives you a value for speed.