2.1.4Physical quantities and kinematics

Gradient of a Displacement-Time Graph

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Lesson summary

2.1.4 Gradient of a Displacement-Time Graph — AS PHYSICS 9702 notes: Find velocity from the gradient of a displacement-time graph, using tangents for curves.A displacement-time graph shows the changing position of an object and its direction relative to the start.The gradient is …

The gradient is Δs ÷ Δt
The steeper the slope
A positive gradient is constant velocity in the positive direction
A horizontal line has zero gradient
A straight line means constant velocity
Use a large triangle

Section 1 of 10Gradient equals velocity

Gradient equals velocity

A displacement-time graph shows the changing position of an object and its direction relative to the start.

The gradient is Δs ÷ Δt

The gradient is Δs ÷ Δt which is the definition of velocity, with unit m s^-1.

The steeper the slope

The steeper the slope the greater the velocity.

A positive gradient is constant velocity in the positive direction

A positive gradient is constant velocity in the positive direction a negative gradient is constant velocity in the negative direction.

A horizontal line has zero gradient

A horizontal line has zero gradient the object is stationary.

Section 2 of 10Straight-line graphs

1A straight line means constant velocity

A straight line means constant velocity the gradient is the same wherever it is measured.

2Use a large triangle

Use a large triangle this gives a more accurate gradient.

3A straight line through the origin

A straight line through the origin constant velocity from the start.

4Two objects on the same axes

Two objects on the same axes the intersection shows when and where they are at the same position (they meet), not when they have the same velocity.

Section 3 of 10Finding velocity at a point on a curve

Place a ruler so that it just touches the curve at the chosen point, with the curve symmetrical about the ruler near the contact.

Draw the tangent extending well beyond the curve.

Draw a large triangle on the tangent line, not on the curve.

Calculate the gradient from the triangle.

Quote the gradient with its unit m s^-1.

Section 4 of 10Meaning of the shape of a curve

Meaning of the shape of a curve

An increasing gradient means increasing velocity

An increasing gradient means increasing velocity the object is accelerating.

A decreasing gradient means decreasing velocity

A decreasing gradient means decreasing velocity the object is decelerating.

A turning point has zero gradient

A turning point has zero gradient an instant of rest.

Velocity changing from positive to negative

Velocity changing from positive to negative means the object has changed direction.

A ball rolling down a slope

A ball rolling down a slope has a gradient that grows steadily and an s = ½at² parabola.

Section 5 of 10Average and instantaneous velocity

The gradient of a chord joining two points gives the average velocity between them.

2As the two points move together

As the two points move together the chord approaches the tangent and the average velocity approaches the instantaneous velocity.

3This is the idea of a limit

This is the idea of a limit v = Δs ÷ Δt for a very small Δt.

4A light gate with a shorter time interval

A light gate with a shorter time interval gives an average speed closer to the instantaneous speed.

Section 6 of 10Using the gradient values

1Velocities at different times can be plotted on a v-t graph

Velocities at different times can be plotted on a v-t graph whose gradient then gives acceleration (see 2.1.5).

2Gradient values with uncertainties

Gradient values with uncertainties are found using best-fit and worst-fit lines (see 1.3.2).

3A car braking to a stop

A car braking to a stop gives a displacement-time curve whose gradient decreases to zero.

4A horizontal velocity-time line

A horizontal velocity-time line becomes a straight diagonal line on the displacement-time graph.

Section 7 of 10Describing a displacement-time graph

Describing a displacement-time graph

Describe the graph section by section

Describe the graph section by section and quote numbers.

Straight section

Straight section constant velocity equal to its gradient.

Horizontal section

Horizontal section the object is at rest at a stated position.

Negative gradient

Negative gradient the object returns toward the start.

Curve

Curve say whether the gradient is increasing or decreasing, so whether the object is speeding up or slowing down.

Section 8 of 10More from the specification

Section 9 of 10Key terms

Term	Meaning
Gradient	Change in displacement divided by change in time, which is velocity.
Tangent	A straight line that touches a curve at one point.
Chord	A straight line joining two points on a curve.
Instantaneous velocity	The gradient of the tangent to a displacement-time graph.
Average velocity	The gradient of the chord between two points.
Stationary	At rest; shown by a horizontal line on an s-t graph.
Turning point	A point of zero gradient where the object is momentarily at rest.
Limit	The value approached as the time interval becomes very small.

Section 10 of 10Exam technique

Drawing the triangle on the curve instead of on the tangent line.

Using the intersection of two s-t lines to mean equal velocities — it means equal positions.

Calling a steep s-t slope a large displacement — it means a large velocity.

Forgetting the unit m s^-1 when reporting a gradient.