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

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