The graph shows the change in velocity of an object over time as the object is subject to a constant force and to a drag force acting in the opposite direction to the constant force. Which of the points 𝐴, 𝐵, 𝐶, 𝐷, and 𝐸 most correctly shows the time at which the object starts to move at its terminal velocity?
Okay, so in this question, we’ve been given a graph that shows velocity against time of a particular object. Now let’s say that this is our object in question. We’ve been told that the object is subject to a constant force and to a drag force acting in the opposite direction to the constant force. In other words, we have two forces acting on the object in opposite directions. One of them is a constant force, and the other is a drag force.
We haven’t been told that the drag force is constant, which makes sense, because the drag force depends on the velocity of the object in question. In other words, the larger the velocity, the larger the drag force trying to oppose the motion of the object. So if we draw on a diagram of our object, then let’s also draw on the forces acting on the object.
Now this is a very generic case. We don’t know if the object is moving horizontally or vertically, and to be honest it doesn’t matter which we choose. The principles are still gonna be the same. So let’s choose a common situation that’s used when we’re discussing terminal velocity.
Let’s assume that the object is falling downwards towards the ground. Well, in that case, the forces acting on the object are firstly the weight of the object, which of course acts in the downward direction because it’s the gravitational force on the object and this is a constant force, and secondly the other force on the object is the drag force. Let’s call this force 𝐹 sub 𝑑 for force sub drag.
Now this drag force is caused by the interaction between the object falling and the air surrounding the object, because as the object falls through the air, the air pushes back against the motion of the object. And the faster the object falls, the larger the value of the drag force. So although we’ve picked a very specific situation where the object is falling, it does fit the description in the question. We’ve got a constant force and a drag force that’s acting in the opposite direction to the constant force.
Now let’s imagine that the object has just started falling. In other words, it started at rest and we’ve just dropped it. Well, in that case, the drag force on the object is going to be very close to zero because the object is not moving, so there’s not going to be any air resistance against the object. This means that there’s a very large net force on the object, the net force of course being the weight of the object. And so the object starts to accelerate or gains velocity in the downward direction.
But as it gains velocity, of course, there starts to be a drag force acting on the object. Initially, the drag force is very small. And as the object gets faster, the drag force gets larger, until eventually the magnitude or size of the drag force is exactly the same as the weight of the object.
Now at this point, the net force on the object is zero because the downward force is exactly canceled out by the drag force. But if the net force on the object is zero, then the object is no longer going to accelerate or gain velocity. This is because of Newton’s first law of motion, which says that objects in motion will stay in motion and objects at rest will stay at rest unless a net force acts on the object.
In this situation, as we’ve already said, there is no net force on the object, so it will continue to travel at the same speed. But if it’s traveling at the same speed, then the drag force is going to remain the same. And so we reach the situation where the two forces are completely balanced and both of them are unchanging, at which point the object continues to move at the same velocity. This velocity is known as terminal velocity, and we can see this behavior on the graph.
We can see that, initially, as the object starts moving, it starts with a velocity of zero. But the velocity’s increasing very very quickly. And as it continues to fall, we can see the effect of the drag force increasing, because even though the velocity of the object is still increasing, the rate at which the velocity increases or, in other words, the gradient of the graph, is decreasing. And of course, the gradient of a velocity–time graph is the acceleration experienced by the object.
So here we can see that there is some acceleration, but eventually we get to a point where the gradient is an absolute flat line. In other words, the object is not accelerating and it has reached a constant velocity, whatever this velocity is. We’ll call this 𝑉 sub 𝑡 for 𝑉 sub terminal because that’s the terminal velocity.
So at what point — 𝐴, 𝐵, 𝐶, 𝐷, or 𝐸 — does the graph basically become a flat line? Well, we can see that, at 𝐴, the graph is definitely not a flat line, and neither is it at 𝐵. The same is true for 𝐶, although it’s nearly there. But the very first point at which the graph is a flat line is the point 𝐷. And so the point 𝐷 most correctly shows the time at which the object starts to move at its terminal velocity. And this is the final answer to our question.