A bat strikes a 0.145-kg baseball. Just before impact, the ball is traveling horizontally to the

step 1 Once again, welcome to a new problem. We're given a ball that's traveling. So this is the inform

A bat strikes a 0.145-kg baseball. Just before impact, the...

Key Concepts

Average Force

Average force is defined as the total impulse (change in momentum) divided by the time interval over which the force acts. This concept provides a useful way to relate short, impulsive forces, such as those in collisions, to the resulting change in an object's velocity.

Time Interval in Impulse Calculations

The time interval during which a force is applied is a key aspect of impulse calculations. Short contact times, such as in collisions, mean that even a small impulse can correspond to a very large average force, emphasizing the transient nature of impacts.

Vector Decomposition

Vector decomposition involves breaking down a given vector into its horizontal and vertical components. This concept is crucial when dealing with objects moving in two dimensions, as it allows separate analysis of the forces and motion along orthogonal directions.

Impulse-Momentum Theorem

This theorem states that the impulse on an object (the product of force and the time interval over which it acts) is equal to the change in momentum of the object. It is essential for analyzing collisions and impacts because it links forces exerted over short times to the resulting motion changes.

Momentum

Momentum is a measure of the quantity of motion an object possesses, defined as the product of its mass and velocity. Its vector nature (having both magnitude and direction) is key in collision problems, as changes in momentum determine the forces involved.

Transcript

00:01 Once again, welcome to a new problem.

00:04 We're given a ball that's traveling.

00:08 So this is the information that's given.

00:11 So we have a ball that's traveling towards the right with a positive velocity of 40 meters per second.

00:21 That's the ball traveling towards the right.

00:25 And then it comes into contact with the bat.

00:31 Okay.

00:31 And once it comes into contact with the bat, it takes off upwards at an angle of 30 degrees.

00:43 So this is the information that's given.

00:46 The mass of the ball happens to be 0 .145 kilograms.

00:52 And the time it takes for the ball to have contact with the bat is 1 .75 milliseconds, which we can change right away to seconds by multiplying by a reciprocal of a thousand and that's going to give us 1 .75 times 10 to the negative 3 seconds so that's how much time the ball and the back are in contact remember the ball is coming from the left moving towards the right and then it gets hit gets hit such that it's going to have component velocity so you know there is it has a velocity in the in the horizontal direction okay which you can call v final x and it also has v final y this one is the initial velocity it's in the x so v initial in the x is 40 and then the initial in the y happens to be zero because the ball is not going up or down so that's the information were given, we're supposed to find the, we want to find the horizontal.

02:18 So this is the average force.

02:19 We want to find the horizontal component, the average force, and we also want to find the vertical component of the average force.

02:33 So, you know, we're calling it v average x and then v average y.

02:39 So looking at this problem, you always have to think about the ball in terms of x and y.

02:46 So the first thing we're going to do is to compute the initial momentum.

02:50 This happens, you know, when the ball is moving towards the right v initial in the x happens to be 40 meters per second.

03:00 So the initial momentum in the x is the mass of the ball times the initial velocity.

03:07 In the x, the mass of the ball happens to be 0 .145 kilograms and the initial velocity in the in the x which is in the positive direction is positive 40 meters per second and so if you plug in those numbers you get that the initial momentum in the x is 5 .8 kilogram meters per second that's the initial momentum in the x remember our target is to get the the vertical and the horizontal components of the average force.

03:44 So if you think of it in terms of a diagram, what you're saying is that you have, you have, you know, once the ball moves this way, and then there's a bat, the bat hits the ball that way.

04:00 And so there is a force, there's an average force that's responsible for that within a specific time.

04:07 That average force has two components has the x component and it also has the y component so that's what we're trying to get so we always have to think about the x and y in this problem but everything starts with the initial stuff so this is the initial momentum in the x remember the ball travels up until it comes to to the bat okay so when he hits the bat it's going to go up that way so there are two components, the negative velocity, the negative final velocity in the x, and also the positive final velocity in the y, because it's going upwards.

04:52 If you look at the problem, they're telling us that this ball, it's traveling upwards.

04:57 So if we want to get the final momentum, the final momentum in the x, will need to find, we'll need to take the mass times the final velocity in the x which is going to be negative because it's pointing in the opposite direction.

05:23 And so this is this is now where you use trigonometry.

05:27 Remember, this angle here is 30 degrees.

05:32 Okay? we can even call it theta for purposes of explanations.

05:36 And then this is a velocity vector.

05:40 This is the final velocity.

05:42 This is the final velocity in the x.

05:44 So you can see using trigonometry, the final velocity in the x over the final velocity is the same as cosine of theta.

05:57 So that means the final velocity in the x has to be the final velocity times cosine theta.

06:07 But it's in that direction.

06:10 So this is going to be negative, final velocity theta.

06:14 So simplifying this problem, we have m times the final velocity in the x, which is negative v -final cosine of 30.

06:24 And then this changes to plug in the numbers 0 .145 kilograms.

06:30 That's the mass times negative, the final velocity, as you can recall.

06:37 If you go back, the final velocity was given us, you know, we have to check, it was given us 52 meters per second.

06:50 So don't forget about the final velocity.

06:52 This final velocity right here, it happens to be v final equal to 52 meters per second.

07:01 So going back, that's the number we're going to plug into the formula.

07:05 So remember there's a negative.

07:08 52 meters per second times cosine of 30 times cosine of 30 or we can see the whole thing as a unit so we have the meters per second at the end okay so we're going to have a meters per second at the end so this times cosine of of 30 meters per second if you multiply those two numbers you get negative negative 12, sorry, you get negative 6 .53 kilogram meters per second, kilogram meters per second.

07:54 So, you know, we have two things, critical pieces of information that we've picked up so far.

08:01 You know, we have this baseball bat or baseball that's going up was hit by a bat.

08:07 We have the initial final velocity in the x.

08:12 We also have final velocity in the y.

08:15 We've computed the initial momentum, the initial momentum in the x going back on this page.

08:26 You can see that the initial momentum in the x happens to be this one right here.

08:32 Okay, at 5 .8 kilogram meters per second.

08:35 That's positive.

08:36 And then we also have the final momentum in the x.

08:45 If you go back, it happens to be negative 6 .53 kilograms meters per second, negative 6 .53 kilogram meters per second...

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