Momentum and Impulse

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December 9th 2009, 07:30 PM
ReneePatt

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 ball’s 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 ball’s momentum.
(ii)Find the average force on the bat.
December 9th 2009, 08:04 PM
mathaddict

Quote:

Originally Posted by ReneePatt

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 ball’s 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 ball’s 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)
December 10th 2009, 06:55 AM
ReneePatt

What is u?

Quote:

Originally Posted by mathaddict

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?
December 10th 2009, 07:19 AM
mathaddict

Quote:

Originally Posted by ReneePatt

What is u? Speed?

v and u are final speed and initial speed respectively
December 10th 2009, 07:45 AM
ReneePatt

Light Bulb

Quote:

Originally Posted by mathaddict

v and u are final speed and initial speed respectively

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