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Math Help - Momentum and Impulse

  1. #1
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    Momentum and Impulse

    It may surprise you to learn that the collision of a baseball and bat lasts only about a thousandth of a second. Here we calculate the average force on the bat during this collision by first computing the change in the ballís momentum.

    The momentum p of an object is the product of its mass m and its velocity v, that is, p=mv. Suppose an object, moving along a straight line, is acted on by a force F=F(t) that is a continuous function of time.

    (a) Show that the change in momentum over a time interval [t_{0},t_{1}] is equal to the integral of F from t_{0} to t_{1}; that is, show that

    p(t_{1})-p(t_{0})=\int_{t_0}^{t_1}F(t) dt

    This integral is called the impulse of the force over the time interval.

    (b) A pitcher throws a 90-mi/h fastball to a batter, who hits a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 s and leaves the bat with velocity 110 mi/h. A baseball weights 5 oz and, in US Customary units, its mass is measured in slugs: m=\frac{w}{g} where g=32 ft/s^2.

    (i)Find the change in the ballís momentum.
    (ii)Find the average force on the bat.
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  2. #2
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    Quote Originally Posted by ReneePatt View Post
    It may surprise you to learn that the collision of a baseball and bat lasts only about a thousandth of a second. Here we calculate the average force on the bat during this collision by first computing the change in the ballís momentum.

    The momentum p of an object is the product of its mass m and its velocity v, that is, p=mv. Suppose an object, moving along a straight line, is acted on by a force F=F(t) that is a continuous function of time.

    (a) Show that the change in momentum over a time interval [t_{0},t_{1}] is equal to the integral of F from t_{0} to t_{1}; that is, show that

    p(t_{1})-p(t_{0})=\int_{t_0}^{t_1}F(t) dt

    This integral is called the impulse of the force over the time interval.

    (b) A pitcher throws a 90-mi/h fastball to a batter, who hits a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 s and leaves the bat with velocity 110 mi/h. A baseball weights 5 oz and, in US Customary units, its mass is measured in slugs: m=\frac{w}{g} where g=32 ft/s^2.

    (i)Find the change in the ballís momentum.
    (ii)Find the average force on the bat.
    HI

    (a) From newton's second law , the rate of change of momentum of a body is directly proportional to the resultant force acting on it and is in the same direction as the resultant force .

    F=\frac{dp}{dt}

    dp=F dt

     <br />
\triangle P=\int^{t_1}_{t_0} F dt<br />

    (b) I am not familiar with the conversion of units here . You can check it from your calculator .

    The change in momentum is , \triangle P = mv-mu

    where m is given , v and u are also given so just substitute appropriately but you have to make sure the units are fixed .

    The average force on the bat would be what you got in (1) divided by time (0.001)
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  3. #3
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    What is u?

    Quote Originally Posted by mathaddict View Post
    The change in momentum is , \triangle P = mv-mu

    where m is given , v and u are also given so just substitute appropriately but you have to make sure the units are fixed .

    What is u? Speed?
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  4. #4
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    Quote Originally Posted by ReneePatt View Post
    What is u? Speed?
    v and u are final speed and initial speed respectively
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  5. #5
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    Light Bulb

    Quote Originally Posted by mathaddict View Post
    v and u are final speed and initial speed respectively

    Oh, okay. I'll see if I can figure it out. Thanks!
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