How do gradients work in physics?

Question

I have a graph, showing velocity against time. The velocity starts at 0 then rises to 10m/s in 2 seconds, stays there for 5 seconds, then slows and stops in another 4 seconds. So the velocity on the graph slopes up, remains horizontal, then slopes back down. How do i find the gradient?

Answer 1

The gradient of the velocity in that motion varies with time, so you have to evaluate the gradients in 3 different periods of time: from 0 second to 2 second, from 2s to 7s, from 7s to 11s.

The gradient of the velocity is defined as the rate of change of velocity over time.

grad v = Δv / Δt

- when 0 < t < 2, gradv = Δv/Δt = (10 - 0) / (2 - 0) = 5 (m/s)/s

- when 2 < t < 7, gradv = Δv/Δt = (10 - 10)/(7 - 2) = 0 (m/s)/s

- when 7 < t <11, gradv = Δv/Δt = (0 - 10)/(11 - 7) = -2.5 (m/s)/s

The gradient of v tells how fast v changes. When v doesn't change at all (second segment of the graph), its gradient is 0. The gradient is positive (first segment) when v increases, negative (third segment) when v decreases.  

Answer 2

Use gradient = rise/run = (final velocity - initial velocity)/(final time - initial time)

Now, from the graph, there are three different gradients - an upward slope, a flat slope and a downward slope - so you need to find all three.  Use the points where it changes slope to give you the final and initial velocities and times.  Note, upward slopes are positive and downward slopes are negative, while flat slopes are equal to zero.