Which color graph represents the path of a particle whose velocity is increasing?
We see we’ve been given a graph that represents displacement against time 𝑡. And so, let’s recall the relationship between displacement and velocity. We know velocity is defined as the rate of change of displacement or position of an object. And so, if we define displacement to be equal to 𝑥, we can say that the velocity must be equal to the rate of change of 𝑥. And when we think about rate of change, we think about the derivative of 𝑥 with respect to 𝑡. But we also know that when we differentiate a function, that tells us the slope of that function.
And so, if we differentiate 𝑥 with respect to 𝑡, we get the slope of 𝑥. So, this tells us that the slope of a displacement–time graph must give us the velocity. And so what we’re really asking ourself in this question is which graph has increasing slope. And in fact, there are a couple of graphs that we can disregard immediately. Let’s look at the yellow plot. We know we can disregard this plot. The displacement, in fact, doesn’t change. And we see that the slope of this graph is equal to zero throughout. So, the velocity is zero, and it certainly isn’t increasing.
Similarly, if we consider the red plot, we see that this has a constant slope. In fact, it has a constant negative slope. And so, the velocity is constant and negative. It’s certainly not increasing. This also helps us to disregard the green plot. It might look like the velocity is increasing, but actually what is increasing is the displacement. This is a straight line, and so its slope is constant throughout. This means whilst the velocity is positive, it’s also constant. And so, it can’t be increasing.
And so we check the blue plot. If we draw tangents to this graph, we see that the slope of the tangents at these points is always positive. So, the velocity is positive. But also, these tangents become more steep. And so, the slope of this graph is increasing throughout. This means the velocity of this particle must also be increasing. And so, the answer is blue. The blue graph represents the path of particle whose velocity is increasing.