Video: Graphically Representing the Horizontal Motion of a Projectile

An object is set in motion by an initial Force 𝐹 that acts diagonally upward as shown in the diagram. The object undergoes projectile motion. Which of the graphs (a), (b), (c), and (d) shows the changes in the horizontal displacement of the object between leaving the ground and returning to the ground? Which of the graphs (e), (f), (g), and (h) shows the changes in the horizontal velocity of the object between leaving the ground and returning to the ground?

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Video Transcript

An object is set in motion by an initial Force 𝐹 that acts diagonally upward as shown in the diagram. The object undergoes projectile motion. Which of the graphs (a), (b), (c), and (d) shows the changes in the horizontal displacement of the object between leaving the ground and returning to the ground?

Okay, so we need to look at graphs a, b, c, and d and determine which one of them is showing the horizontal displacement of this object here in the diagram. Now the object starts out at this position here initially and it moves along the dotted trajectory that has been given to us in the diagram. And if we pick a random point in this trajectory, let’s say this point here, then at that point we can work out the object’s horizontal and vertical displacement relative to its starting point. Now in this case, the object’s vertical displacement is this distance here because that’s how much further above its initial point the object has moved. And its horizontal displacement is this distance here because that’s how much horizontally the object has moved.

For this question though, we’re only interested in the horizontal displacement. So let’s get rid of the arrow describing the vertical displacement. Now as the object moves along its trajectory, we can see that the horizontal displacement of the object is increasing. In other words, it’s getting further and further away from its starting point. And this is true for the entire trajectory of the object. So immediately, we can rule out graph d because graph d is suggesting to us that as time goes on, the horizontal displacement of the object stays constant; it stays the same. But this is definitely not the case, the horizontal displacement is increasing over time.

More importantly though, how is it increasing overtime? Is it increasing as in graph a, graph b, or graph c? Now graph a is suggesting that horizontal displacement increases very quickly, then stops increasing, and then start increasing very quickly once again. Graph b is suggesting that the horizontal displacement starts out increasing very slowly, then suddenly increases very rapidly in the middle, and then slowly increases again. And graph c is suggesting that the horizontal displacement increases constantly as time progresses. So which one is correct?

Well, to answer this, we need to realize that the object has some velocity of over its entire trajectory. And at every point along its trajectory, we can split up its velocity into the horizontal and vertical components. For example, at this point over here, we can see that the object is moving in this direction. And we can split that velocity into a horizontal component and a vertical component. Similarly, at this point, the object is moving in this direction, which is mainly horizontal. So it’s gonna have a horizontal component, and its vertical component is gonna be fairly small. Now at this point here, the object is only moving horizontally. So it only has a horizontal component, and so on and so forth. We can do this for all the other points of the object’s trajectory.

However, the important thing to note is that once we’ve exerted force 𝐹 onto the object, the object starts moving along its dotted trajectory. And at that point, the only force acting on the object is the gravitational force. Let’s call that force 𝑤 for the weight of the object. And the weight of the object is constant and always acting downwards. Now when we split the object’s velocity into the horizontal and vertical components, we can therefore say that the weight only acts on the vertical component of the object’s velocity because the weight is going to result in an acceleration of the object in the downward direction, or in other words in the direction of the force. This means though that the horizontal component of the object velocity remains unchanged because there is no force acting on the object in that direction. And hence the object will not accelerate or decelerate in horizontal direction.

Therefore, at every point along its trajectory, the object’s horizontal velocity is constant. And at this point, we can recall the equation speed is equal to distance divided by time. Or in this particular case, the horizontal velocity of the object is equal to the distance travelled by the object in the horizontal direction divided by time. Now the distance travelled in the horizontal direction is the same as the horizontal displacement of the object. And as we’ve just said, the horizontal velocity of the object remains constant. And so as time progresses, the horizontal displacement of the object must also increase at a constant rate. In other words then, the graph showing the correct horizontal displacement for our object is graph c. And that is our final answer to this part of the question.

So now that we’ve considered these four graphs which talked about horizontal displacement on the vertical axis, let’s talk about these four graphs which consider horizontal velocity on the vertical axis. But hang on! In order to answer the first part of the question, we’ve already considered horizontal velocity. Amazing! This means we’ve made life easier for ourselves. So let’s rejoice in this fact and look at the next part of the question.

Which of the graphs e, f, g, and h shows the changes in the horizontal velocity of the object between leaving the ground and returning to the ground?

Okay so once again, we can consider the object at different points along its trajectory. And yet again, we can find the horizontal and vertical components of this object’s velocity. So at this point here, we’ve got a horizontal component and some vertical component as well. At the second point, we’ve got the same horizontal component as before, but a slightly smaller vertical component. Now remember, the horizontal component stays the same because earlier we said that there are no forces acting on the object in the horizontal direction. Therefore, there is no acceleration or deceleration of the object in the horizontal direction.

Similarly, at the third point, we’ve got the horizontal component, but this times the object is only moving towards the right. So there is no vertical component. At the fourth point we have the same horizontal component once again, but this time we have a downward vertical component that’s fairly small. And finally at the fifth point, we have the same horizontal component, but a larger downward vertical component. And we can do this for every point along the object’s trajectory. But the point is that the horizontal velocity of the object does not change. And so on a graph of horizontal velocity against time, we’re looking for the graph that shows a constant horizontal velocity.

So that’s this graph here because graph f suggests that the horizontal velocity increases and then decreases, graph g suggests that the horizontal velocity keeps increasing, and graph h suggests that the horizontal velocity increases in some weird manner. And so we’ve found the answer to this part of the question as well. Graph e correctly shows the changes in the horizontal velocity of the object between leaving the ground and returning to the ground in that it correctly shows us that the horizontal velocity doesn’t change at all.

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