Video Transcript
The diagram shows the changes in the horizontal and vertical displacement of an object at equal time intervals. Gravity is the only force acting on the object. Is the object’s horizontal speed increasing, decreasing, or constant?
In our diagram, we’re shown the position of our object at six different moments in time. We’re told that each of these time intervals is the same. In other words, the amount of time that passes from 𝑡 zero to 𝑡 one is the same as the amount of time that passes from 𝑡 one to 𝑡 two and from 𝑡 two to 𝑡 three, and so on. At these six instants in time, we see both the horizontal as well as the vertical position of our object as it goes through projectile motion. We know it’s experiencing this kind of motion because gravity is the only force acting on it.
The first part of our question asks if the object’s horizontal speed is increasing, decreasing, or constant. We can recall that, in general, the speed of an object equals the distance it travels divided by the time taken to travel that distance. Considering horizontal motion, we know that over the first time interval, our object travels this distance. Over the second time interval, it travels this distance, the same as the first. Over the third interval, it travels this distance, which is also equal to the first two, and so on over all five equal intervals of time. Since over these equal time intervals the object is always traveling the same distance, that means its speed in this direction is constant. Our answer to the first part of our question is that the object’s horizontal speed is constant.
Let’s look now at part two.
This part of our question asks, is the object’s vertical speed increasing, decreasing, or constant?
Like before, we’ll look at the distances traveled by our object over the intervals of time given. Unlike before, now, we’ll consider the vertical distances. Over the first time interval, from 𝑡 zero to 𝑡 one, the vertical distance traveled is given by this arrow here. Then from 𝑡 one to 𝑡 two, the vertical distance traveled is given by this arrow, then this arrow, then this one, and finally this one over the last time interval. Because the lengths of these arrows are increasing as time increases, we know that over each successive interval, there’s more distance being covered. This tells us that in the vertical direction, the object is speeding up. And this agrees with our practical experience. If we release a ball from rest out of our hand, we know that as the ball falls, it speeds up. All this to say the vertical speed of our object over these intervals of time is increasing.
Let’s look now at the next part of our question.
Is the object’s horizontal acceleration increasing, decreasing, or constant?
To begin, let’s recall that the acceleration of an object in general is equal to that object’s change in speed, Δ𝑠, divided by the amount of time passed, Δ𝑡. This part of our question actually relates to previous parts we’ve answered. For example, recall that we found the horizontal speed of our object to be constant over this interval of time from 𝑡 zero to 𝑡 five. Concerning the object’s horizontal acceleration then, that means that Δ𝑠 is zero. There is no change in the object’s horizontal speed over time. Therefore, we can say our object’s horizontal acceleration is zero. In terms of our answer though, we’ll write that it is constant. Zero after all is a constant value. So, in the horizontal direction, our object’s acceleration is constant.
Let’s consider now the next part of our question.
Is the object’s vertical acceleration increasing, decreasing, or constant?
Regarding our object’s vertical motion, we know that it is changing speed over time. That is, it has some nonzero acceleration. We know this because the length of these lines indicating the distance traveled in each interval of time increases as time increases. But simply knowing that this object’s vertical acceleration is nonzero doesn’t tell us whether it’s increasing, decreasing, or constant. To work that out, we can return to our question’s statement, where we read that gravity is the only force acting on the object.
Gravity is a force that creates an acceleration. Over small vertical distances, we treat this acceleration as effectively constant. That is, we typically represent the magnitude of the acceleration due to gravity as 9.8 meters per second squared. So, this then is the vertical acceleration of our object. And note that this is a constant value. In the vertical direction, our object does speed up, but it does so at a constant rate, the acceleration due to gravity. So, in the vertical direction, just like in the horizontal, we have a constant acceleration.
Let’s look now at the last part of our question.
Is the object’s total speed increasing, decreasing, or constant?
To get started, let’s recall that the horizontal speed of our object maintains a constant value, while the vertical speed we found earlier to be increasing. The object’s total speed takes into account both its vertical and its horizontal speeds. If we call 𝑠 sub v the object’s speed in the vertical direction and 𝑠 sub h its speed in the horizontal direction, then the object’s total speed — we’ll call it 𝑠 — equals the square root of the sum of the squares of 𝑠 sub v and 𝑠 sub h. In other words, the total speed of our object depends on its vertical and horizontal speeds. The horizontal speed, we said, is constant, while the vertical speed is increasing. Therefore, the overall value of 𝑠, that is, the object’s total speed, is increasing as well. In answer to this part of our question then, we say that the object’s total speed is increasing over time.
