This video will illustrate how to use

the ideas of impulse and momentum to solve a problem. The first part of the

problem states that a tennis player tosses a tennis ball with a mass of 58.5

grams up in the air to serve it.She strikes it at the moment it stops moving

upward, and it speeds away at seventy-seven point five miles per hour.

We are asked what is the ball’s velocity in meters per second when it leaves the

racket? If the ball is in contact with the racket for 3.7 six milliseconds

what force does the ball feel well the first step in the problem-solving

process is always to read the problem carefully and pick out the data that

we’re given and in this problem we are given the mass of the ball at 58 point

five grams we know that the initial velocity of the ball is going to be zero

because she strikes it at the moment it stops moving the final velocity of the

ball when it leaves the racket is seventy seven point five miles per hour

and the time of contact is three point seven six milliseconds now one thing we

notice here right off the bat is that pretty much all of our data is in the

incorrect units and so we need to convert these units immediately before

we do any further calculations so we convert the mass of the ball from grams

into kilograms using the fact that there are 1000 grams in one kilogram this

gives a mass for the ball of 0.058 five kilograms we must convert the velocity

into meters per second as we’re asked in the first question we use the fact that

there are 1609 meters in one mile and the fact that one hour has 3600 seconds

this gives us a final velocity for the ball of 34.6

meters-per-second then finally we use the fact that one second has a thousand

milliseconds to convert the time of contact to zero point zero zero three

seventy six seconds so we have picked out the data from the problem and we

have put them in the proper units now we need in the second step to figure out

what the problem is asking for and in the second problem here we are asked the

force that the ball feels and so we make a notation here on our little data table

that we do not know the force then we move on to the third step which is to

relate what you are told to what you need to figure out and in this case we

need to use the formula for impulse which states that impulse is equal to

the change in momentum in other words force multiplied by time is equal to the

final momentum minus the initial momentum so force multiplied by time is

going to be the mass of the ball times the final velocity minus the mass of the

ball times the initial velocity we can immediately simplify the equation by

realizing the initial velocity is zero so the initial momentum is zero and so

we’re left with the force multiplied by the time zero point zero zero three

seven six seconds is equal to the mass of the ball zero point zero five eight

five kilograms multiplied by the final velocity of the ball thirty-four point

six meters per second solving the equation for F we get that

the force is two point zero two four kilogram meters per second notice the

odd units for momentum divided by zero point zero zero three seven six seconds

that gives us a force 555 Newtons two three significant figures notice the

units here kilogram meters per second divided by seconds becomes kilogram meters per second squared which is a

Newton going to the second part of the problem we say air resistance reduces

the ball speed to sixty-two point seven miles per hour by the time it reaches

the opponent the opponent strikes the ball and sends it back along the same

line at seventy point one miles per hour this time the ball is in contact with

the racquet for four point two five milliseconds and we’re asked how much

force does the ball feel this time well some of our data have changed the mass

of the ball remains the same of course the mass of the ball is still zero point

zero five eight five kilograms but now the initial velocity has become sixty

two point seven miles per hour and the final velocity is negative seventy point

one miles per hour notice we have assigned a negative sign to the final

velocity because V naught is going to be toward the tennis player and then the

final is going to be away from the tennis player velocity is a vector

quantity just like momentum it has direction and we tell that direction by

saying the incoming velocity is positive than the outgoing velocity is then going to

negative doing our unit conversion with these data we can change the miles into

meters and the hours into seconds so we discover that the initial velocity is

twenty eight point zero meters per second to three significant figures and

the outgoing velocity converting by a similar process we discover to be

negative 31 point three meters per second putting the negative sign there

recognizing that the outgoing velocity is in the opposite direction of the

incoming velocity is a vital part of this problem so now we can calculate the

initial velocity by taking the mass of the ball times the initial velocity to

get the initial momentum that’s zero point zero five eight five times twenty

eight twenty eight point zero and that gives us an initial velocity of one

point six three eight kilogram meters per second similarly the final velocity

is the mass of the ball times the final velocity which is zero point zero five

eight five times negative 31 point three meters per second which gives us a final

momentum of negative one point eight three one kilogram meters per second

again the negative sign indicating that the final momentum is in the opposite

direction of the initial momentum now we calculate our change in momentum which

is P final minus P initial that’s negative one point eight three one

– one point six three eight that gives us a change in momentum of negative

three point four six nine kilogram meters per second this is why keeping

track of the signs is very important so returning to our impulse equation force

times time equals change in momentum that gives us force times time is

negative three point four six nine with a new time of four point two five

milliseconds we now can calculate the force as being three point four six nine

kilogram meters per second divided by zero point zero zero four two five

seconds the new time of impact and that gives us an impact force of 816 Newtons

notice the tennis ball feels a lot more force on this return shot than it did on

the serve because in addition to changing the size of the momentum we

also changed the direction of the momentum changing both the size and the

direction of momentum requires a much greater force let me also note that

these are the forces felt by the ball the force exerted by the tennis player

in both cases is going to be higher than this because the tennis player is not

only striking the ball the tennis player also has to move the racket and moving

the racket requires force as well so the actual force exerted by the tennis

player is going to be greater in both cases these are the forces felt

the tennis ball