Slope gives acceleration

The idea

The slope of a velocity-time graph gives the acceleration, a = (v – u)/(t2 – t1); a negative slope means the acceleration is opposite to the direction of velocity.

Steep or gentle

On a velocity-time graph, one car's line climbs steeply and another's climbs gently.

What do you think a steeper climb on this graph means?

• The velocity is changing faster
• The car is heavier
• The car has travelled a longer road

Move the two points

Move the velocities at 10 s and 20 s and read the slope.

Rise over run again

From 5 m s⁻¹ at 10 s to 10 m s⁻¹ at 20 s the slope was (10 − 5) ÷ (20 − 10) = 0.5 m s⁻². Falling from 10 to 5 gave −0.5 m s⁻². The slope is the rate of change of velocity, which is the acceleration.

The slope of a velocity-time graph gives the acceleration: a = (v − u)/(t2 − t1). A negative slope means the acceleration is opposite to the direction of velocity.

Notes

The slope of a velocity-time graph is the acceleration; a negative slope means the acceleration is opposite to the velocity.

Check yourself

The slope of a velocity-time graph gives…

A velocity-time line passes through 5 m s⁻¹ at 10 s and 10 m s⁻¹ at 20 s. What is the acceleration?

Answer: 0.5 m s⁻²

(10 − 5) m s⁻¹ ÷ (20 − 10) s = 0.5 m s⁻².

A velocity-time line passes through 10 m s⁻¹ at 10 s and 5 m s⁻¹ at 20 s. What is the acceleration?

Answer: -0.5 m s⁻²

(5 − 10) m s⁻¹ ÷ 10 s = −0.5 m s⁻².

A velocity-time graph slopes downward. This means the acceleration is…

• the acceleration — correct. Yes!
• the velocity. The velocity is read from the height of the graph, not from its slope.
• the displacement. The displacement is found from the area between the graph and the time axis, not from the slope.
• in the direction of the velocity. That happens when the graph slopes upward and the velocity increases.
• opposite to the direction of the velocity — correct. Yes!
• zero. A level line has zero acceleration. A sloping line does not.