Bill can throw a ball vertically at a speed 1.5 times faster t...

Question

Bill can throw a ball vertically at a speed 1.5 times faster than Joe can. How many times higher will Bill's ball go than Joe's?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Tom is able to fire his arrow vertically with a velocity that is 1.80 times greater than the velocity of Alex's arrow. Determine how much higher Tom's arrow will go up into the air compared to Alex's arrow when they are both fired vertically from the same starting point. So that's her end goals. We're trying to figure out how high Tom's arrow will go when compared to Alex's arrow, if they're both fired or shot in the air at the same starting point. Fantastic. So we're also given some multiple choice answers. They're all in the same units of meters. So let's read them off to see what our final answer might be. A is 3.60 B is 1.80 C is 3.24 and D is 0.90. Awesome. So first off, let us consider the upwards motion to be positive and that Y I is equal to 0 m. So this is the initial distances at the ground. We'll make a little note of that. Awesome. So this is the ground level where the arrows are shot. OK. And that is why the initial, so Y I the initial distance. OK. So now we can recall and use our U AM which is our uniformly accelerated motion equations. So let us start by focusing on Tom's arrow first. So the final velocity of Tom's arrow squared is equal to the initial velocity of Tom's arrow squared plus two multiplied by the acceleration of gravity. Awesome multiplied by delta Y one. And let's note that delta Y one is equal to Y one minus Y zero, which is equal to Y one minus 0 m, which is equal to just Y one. And that our acceleration due to gravity is represented by a subscript G and this case will be a negative 9.81 m because gravity is trying to pull these arrows back down to the ground. And then our final velocity is equal to 0 m per second because when these arrows are shot from the ground strap in the air. So when it hits the highest point right before it starts to fall back to earth. So that's the max height. So for a split second, when it's up in the air before it starts to fall back down, it's 0 m per second. Awesome. So now we need to rearrange this equation to solve for the initial velocity. And then we can plug in all of our known variables to help us solve for our initial velocity. And this is focusing on Tom's arrow. OK. So let's do that. So the final velocity we just discussed is 0 m per second. So 0 m per second squared is equal to the initial velocity of Tom's arrow squared plus two multiplied by the acceleration due to gravity which is negative 9.81 m per second squared multiplied by or why? One value? Awesome. Awesome. So let me simplify and get our initial velocity onto one side isolating itself for initial velocity. We'll call it equation one. We'll get that the initial velocity for Tom zero squared is equal to two multiplied by gravity which is negative 9.81 m per second squared multiplied by Y one. Awesome. So now we need to solve for Alex's arrow. So using the same formula format, we'll get that the final velocity for Alex's arrow squared is equal to the initial velocity of Alex's arrow squared plus two multiplied by the acceleration due to gravity multiplied by delta Y two. OK. And like before you know that delta Y two is equal to Y two minus Y zero, which is equal to Y two minus 0 m is equal to Y two. And then the final velocity of Alex's arrow is also 0 m per second at the maximum height. And we know the initial velocity of outlets zero is the value that we're trying to solve for. That's what we know we need to solve. Four. Awesome. And then the acceleration due to gravity. Awesome. Awesome. Which is a G A subscript G. OK. So like before we need to plug in all of our known variables and solve for the initial velocity of Alex's arrow. So we will get that the initial velocity of Alex's arrow squared is equal to two multiplied by 9.81 m per second squared multiplied by Y two. And let's call this equation two. However, we must note that the initial velocity of Tom's arrow is 1.8 times greater than Alex's arrow. So we must write that the initial velocity of Tom's arrow is equal to 1.8 zero. Since in the problem, it's given to us as 1.80 times multiplied by the final velocity of Alex's arrow. OK. So now we must square both sides. OK? So we'll get that the initial velocity of Tom's row squared is equal to 1.80 squared multiplied by the final velocity of Alex's arrow squared. Awesome. So now we need to plug in all of our known variables and solve for Y one which is the final distance of Tom's arrow, which is the final answer that we're trying to solve. That's gonna be one of our multiple choice answers. OK? So let's do that. So two multiplied by gravity 9.81 m per second squared multiplied by Y one is equal to 1.80 squared, multiplied by two, multiplied by 9.81 m per second squared multiplied by Y two. So therefore, when we isolate and solve for Y one, we will get that Y one is equal to 3.24 multiplied by Y two. Thus, Tom's arrow will go 3.24 times higher then Alex's arrow. So therefore our correct answer as to be the letter C 3.24 m. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Bill can throw a ball vertically at a speed 1.5 times faster than Joe can. How many times higher will Bill's ball go than Joe's?

Key Concepts

Kinematics

Projectile Motion

Energy Conservation

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