A ball rolls off a 300 meter cliff with a horizontal speed of 20 meters per second and hits the ground. How long does it take to hit the ground? Feel free to try this problem if you want to.
But let's begin by drawing a picture. So we have a ball. It rolls off horizontally from a cliff, and then it's going to fall and hit the ground. Let's call this point A and point B.
Now, we know the initial speed. The initial horizontal speed is 20 meters per second. We know the height of the cliff.
It's 300 meters. With this information, how can we calculate the time of flight? How long it takes for the ball to hit the ground at point B? Well, if you know the height of the cliff, you could use this formula. h is equal to 1 half gt squared.
If you rearrange that formula and solve for t, you get that t is equal to the square root of 2 times the height over g. By the way, you can find these formulas all in my formula sheet down in the description section below. So I have this formula directly for this specific type of trajectory. So t is going to be the square root of 2 times 300, the height of the cliff, divided by g, which is 9.8. So the time it takes to go from point A to point B is 7.8.
246 seconds. So that's the answer for part A. Now let's move on to part B.
What is the final velocity of the ball just before it hits the ground? So we're not looking for final speed, but final velocity, which means we're looking for a vector. We need the magnitude and the direction. So let's start with the magnitude.
We could find the final velocity using this formula. It's v initial squared minus 2gt v initial sine theta plus gt squared. With this formula, as long as you know the initial speed. the launch angle, and the time it takes for the ball to hit the ground, T, you could find the final velocity when it hits the ground.
In fact, you could find the final velocity at any point, as long as you know the time and if you have the initial velocity and the initial launch angle. Now, what you need to know is that at point A, the initial angle is 0 degrees because the ball is moving horizontally. And sine of 0 degrees is 0, so this entire term...
So for this specific trajectory, this formula simplifies to this one. V final is equal to V initial squared plus GT squared. V initial is 20. G is 9.8.
And the time we have that is 7.8246. Don't forget to square this. So 9.8 times 7.8246, that's 76.68. When you square that and then add 20 squared, you get 6280. If you take the square root of that, you get the final speed of 79.246 meters per second. So this is the magnitude.
In order to get the velocity vector, we need the angle. So at point B, the ball has an x-component. and it has a y component and we're looking for VF and the angle. Now if we know VY and VX we could use arctangent but right now we have VX and VF so arccosine is going to be more helpful. So the angle is going to equal arccosine Vx over Vf.
And we already know Vx. At point A, V initial is equal to Vx because the ball is moving completely in the horizontal direction. It doesn't have any y component at point A, or at least the y component is 0. So it's going to be arc cosine Vx is 20 over Vf, which is 79.246.
So this gives you an angle below the x-axis or below the horizontal of 75.38 degrees. So that's this angle below the x-axis. Now there are some other ways in which you can get the answer.
So Vx is going to be 20 meters per second at point A and point B. Vx doesn't change because there's no acceleration in the horizontal direction, in the x direction. We do have gravitational acceleration in the y direction.
changes. Vy is going to be different at point B than it is at point A. So to get the same answer using an alternative method, you need to find Vy. And you can use this formula.
Vy final is equal to Vy initial, which is V initial sine theta minus gt. V initial is 20. Well for this particular trajectory, the angle, the initial angle is zero, so sine zero is zero, this part disappears. So in a formula sheet for this particular trajectory, I have that Vy final, it's simply negative GT. You can find that in the formula sheet down below. So it's going to be negative 9.8 times 7.8246.
And this gives you a Vy final of negative 76.68 meters per second. Now that we have Vy final, we can find V final using this formula. So it's going to be the square root of 20 squared plus, once we square it, the negative sign won't be important.
And this will give you the same answer of 79.246 meters per second. I actually used the exact answer here. So as you can see, we got the same magnitude.
Now let's calculate the angle. So here's Vx. At point B, this is Vy, or Vy final, and this is V. So since we know Vx and Vy, we could use arctangent instead of arccosine.
So arctangent is going to be opposite over adjacent, opposite to the angles Vy, adjacent to is Vx. Arccosine is adjacent over hypotenuse, hypotenuse being Vf. So hopefully you remember your trig, SOHCAHTOA. With that, you could use arccosine or arctangent to find the angle based on what you have. So, VY, I'm going to use the positive value for VY, 79.246, and VX is 20. And I get the same answer.
I got 75.8 degrees, but that's close to 75.38. Actually, I see my mistake. I used the wrong value. This should be 76.68 over 20. I mixed up those two numbers.
So now this gives me the same answer, 75.38 degrees. So that's how you can calculate the final velocity. That first formula was, as you can see, a lot easier to use. It saved you a lot of time.
But you can also do it the traditional way, if you don't know that formula. Number two, a ball is kicked from the ground at 30 degrees with an initial speed of 50 meters per second. Calculate the time of flight. Let's begin with a picture. So it's kicked from the ground.
which means it's going to go up and then it's going to go down we got three points of interest point a point b point c they could be other points but those are like the main three Now, we know that it's kicked with a speed, the initial, of 50 meters per second, and the launch angle is 30 degrees. Calculate the time of flight, so that is the time it takes to go from A to C. In the formula sheet down below, you'll find that the time of flight for this specific trajectory is 2V sine theta, this is V initial, over G. So we just have to plug everything into that formula to get the answer. So it's 2 times 50 times sine 30 over 9.8.
Sine 30 is 1 half, 1 half times 2 is 1, so these cancel, which means it's 50 over 9.8 for this particular problem. So the time of flight from point A to point C is 5.102 seconds. Now, part B.
What is the final velocity of the ball just before it hits the ground? Notice that point A and point C, they have the same vertical position. That means that the final velocity at point C is the same as point A. The only difference is the direction. So at point C, the final velocity will be 50 meters per second at an angle of 30 degrees.
Here, this angle is above the horizontal, but here it's going to be... 30 degrees below the horizontal. That's the difference.
So the direction is slightly different, because instead of going up, you're going down. But the magnitude will be the same. Now, we can confirm this answer using this formula again. v final is equal to the square root of v initial squared minus 2. gt v initial sine theta plus gt squared. Let's use that formula to confirm our answer.
So v initial is 50 minus 2 times g times t, which To save space, let's write 5.1 times V initial times sine of the angle, that's sine 30, plus gt. g is 9.8, t is, alright we got space, 5.102. Don't forget this negative sign here.
So square root 50 squared minus 2 times 9.8 times 5.102 times 50 times sine 30. Plus, don't forget the parentheses here, 9.8 times 5.102. Close parentheses and square it. So I got 49.9998, which is approximately 50 meters per second. So as you can see, you'll get the same answer.
And if you calculate the angle, it will be the same as well. Number three, a ball is kicked at an angle of 30 degrees from a 400 meter cliff with an initial speed of 80 meters per second. How long will it take the ball to hit the ground? In other words, what is the time of flight? So like always, let's draw a picture.
So this is going to be the third case. So here we have a ball. It's kicked from a cliff. It's going to go up, and then it's going to go down. So there's three points of interest, point A, point B, and point C.
So we know the height of the cliff is 400 meters. We also know the launch angle. It's 30 degrees. And we know the initial speed is 80 meters per second. So that's V initial.
If you have the formula sheet, you'll find that we could use this formula to find the time it takes to hit the ground. So the time it takes to go from point A to point C, the time of flight, it's the initial sine theta plus the square root. V initial squared sine squared theta plus 2G y initial. Y initial is the height of the cliff over G.
So this equation comes from the quadratic equation when you set it up. Let's go ahead and plug everything in. So the initial is 80. We have the launch angle of 30 plus square root 80 squared sine 30 squared.
You just have to make sure you plug in everything correctly. And then we have plus 2 times 9.8 times the height of the cliff, which is 400. and then all divided by g. So, might take me a minute to plug this in.
The answer I got is about 14 seconds. 13.996 seconds. Now, let's move on to Part B. What is the final velocity of the ball just before it hits the ground?
To calculate that, we could use the formula we've been using. It's this equation. And you could find it in the formula sheet down below. This time, we need to use everything in this formula.
So V initial is 80. So we got 80 squared minus 2 times 9.8. And then T, I mean, we could just put 14 at this point. It's pretty close to 14. And then times V initial times sine of 30 degrees plus GT squared. So 9.8 times 14 squared.
So V final, I got 119.36 meters per second. So that's the magnitude. Now to find the angle below the horizontal, it's arc cosine Vx over V final. We don't know Vy final, so we're not going to use arc tangent. Now, Vx is, if you remember, it's a V initial cosine theta, where theta is the initial angle at point A.
And then if we divide it over V final, it will give us the angle. So this is arc cosine. V initial is 80, and then times cosine of 30 degrees over V final, which is 119.36.
So the angle that I got is 54.5 degrees and this is below the horizontal or below the x-axis because it's going towards quadrant 4. So if the problem... simply asks for the final speed, we would just report this answer. But because we want the final velocity, we need to report the magnitude and the direction.
So those two combined will give us the final velocity vector. So that's it for this problem. Without these formulas, doing a problem like this will take a lot longer.
As you can see, we were able to finish this problem under five minutes. So that formula sheet that I have down below will be very helpful for difficult problems like this.