Video Transcript
A child kicks a ball and it moves along a horizontal ground towards a wall two meters away. The ball moves at 2.25 meters per second toward the wall, hits the wall, and bounces half the distance straight back to the child, but moving only at an average speed of 1.75 meters per second after hitting the wall. What total distance did the ball move? What was the ball’s net displacement towards the wall? How much time did the ball move for? Give your answer to two decimal places.
And then there are two further parts to this question that ask us to calculate the speed and velocity of the ball, respectively. So, let’s begin by calculating the total distance that the ball moved. We’re first told that the child kicks the ball and it moves towards a wall two meters away. It then rebounds half the distance back towards the child. Well, half of two is one, so the ball travels a further one meter back towards the child. The distance tells us how far the ball has traveled altogether, so direction is unimportant here. The distance is simply two meters plus one meter, which is equal to three meters. This means the total distance that the ball moved was three meters.
The second part of this question wants us to calculate the net displacement toward the wall. So, what is the difference between distance and displacement? We said that when calculating distance, the direction doesn’t matter. With displacement, it does. Let’s go back to our earlier diagrams. And since we’re calculating displacement toward the wall, let’s define the direction when the ball is moving towards the wall to be positive. This means when the ball is traveling in the opposite direction, its displacement is said to be negative. And so the total displacement is two minus one, which is equal to one. And so the net displacement of the ball is one meter.
The third part of this question wants us to calculate the time that the ball was moving for. Now, we are given the speed of the ball over the two periods of its motion. And of course since we have the distance that the ball travels during each part of its motion and its speed, we can calculate its time. We’ll use the formula speed is distance over time. Remember, just as with distance, speed does not have a direction, whereas velocity does.
If we define 𝑡 sub one to be the time taken for the ball to travel from the child to the wall, we can say that 2.25, that’s the speed, is equal to two, that’s the distance, divided by the time. Since we’re working in meters per second and meters, respectively, the time is going to be in seconds. Then, we can rearrange for 𝑡 sub one by multiplying both sides by 𝑡 sub one and then dividing by 2.25. 𝑡 sub one is therefore two divided by 2.25, which is eight-ninths, or eight-ninths of a second.
Let’s repeat this process for the second part of the journey. This time, the ball traveled just a distance of one meter, and it did so at a speed of 1.75 meters per second. So, our formula is 1.75 equals one divided by 𝑡 sub two. Once again, we’ll rearrange for 𝑡 sub two. And we do so by multiplying by the variable and then dividing by 1.75. So, 𝑡 sub two is one divided by 1.75, which is equal to four-sevenths. So, the ball took four-sevenths of a second to travel in the direction away from the wall. The total time over which the ball was moving is therefore the sum of these. It’s eight-ninths plus four-sevenths, and that’s 1.4603 and so on. Correct to two decimal places, that’s 1.46. So, the ball was traveling for a total of 1.46 seconds.
Now that we’ve answered the first three parts of this question, we’ll answer part four and five. We’ll need to clear some space to do so. But we’re going to keep some of the information on the screen.
The fourth part of this question asks us, “What was the ball’s average speed during its motion to the nearest meter per second?”
Now, remember, we said that speed is nondirectional. It’s simply the total distance traveled divided by the total time. In this case then, the average speed is three divided by 1.46. Three is the total distance, whereas the time is 1.46 seconds. Three divided by 1.46 is 2.054 and so on. Correct to the nearest meter per second, this is equal to two. So the average speed during its motion was two meters per second.
What was the ball’s average velocity toward the wall from when the ball was kicked to when it had rolled back from the wall and come to a stop? Give your answer to two decimal places.
To calculate the velocity, we remind ourselves that of course velocity is directional. This means that we can calculate the velocity of the ball by considering its displacement. Velocity is displacement divided by time. In this case, we said that the displacement of the ball was one meter, so the velocity is one divided by 1.46. That’s 0.684 and so on. Correct to two decimal places, that’s 0.68. Remember, earlier on, we defined the direction towards the wall to be positive. So the average velocity towards the wall is 0.68 meters per second.
