Energy and Momentum: The Ballistic Pendulum

Transcription

1 Physics Departent Handout -10 Energy and Moentu: The Ballistic Pendulu The ballistic pendulu, first described in the id-eighteenth century, applies principles of echanics to the proble of easuring the velocity of a bullet. As a device in practical ballistics it is now obsolete, but the echanical theory on which it is based, which involves both oentu and energy conservation principles, and an understanding of when echanical energy is not conserved, akes a laboratory study of the odel valuable. In its original for, the ballistic pendulu consisted of a heavy block, suspended by long cords, into which a bullet was fired. The subsequent recoil otion of the block was analyzed to obtain the speed of the bullet. The odel of the apparatus that we will use fires a etal dart fro a spring-loaded launcher. The dart has a suction cup that causes it to stick to a suspended etal block at which it has been fired. The odel preserves the essential principles of the original device. In the collision of the dart with the pendulu we will observe the consequences of oentu and energy conservation. This write up has five ain sections: Section A describes the apparatus and points out the different ways that it can be varied. These variations ay siply be in ass, or ay be in the physics of the firing (recoilless or with recoil of the launcher). Section B describes how the energy iparted to the block is easured. This ethod is the sae no atter how the dart was originally fired. Section C describes how the uzzle velocity of the dart is easured. This easureent ust be ade also, regardless of the different uses that are ade of this velocity in the next two sections. Section D describes the equations of the pendulu experient for a recoilless firing (dart gun fixed) and its procedures. Section E describes the equations of the experient for a firing with recoil (dart gun suspended) and its procedures. Section A - the Pendulu Apparatus A rod about four feet above a leveling base holds an aluinu frae securely. A set of long strings fro the frae support a etal shape that plays the part of the block into which projectiles were fired. In this odel, the block presents a flat surface to a dart fired at it horizontally, and the two will

2 Physics Departent Handout -10 stick together because the etal dart has a suction cup at its tip. There are long screws protruding fro the sides of the block that will support standard laboratory weights. This allows for varying the ass of the target block. A siilar set of long strings fro the frae can support the dart launcher, which also has long screws to allow for the addition of weights to vary its ass. The dart launcher holds the dart gun securely, but the gun can easily be reoved fro the launcher to be used separately. With this arrangeent, the gun can be fired fro the launcher while the latter recoils, or by reoving it fro the launcher, the gun can be fired fro a rigidly ounted position. The dart gun itself has a trigger echanis that releases the dart with a fairly constant uzzle velocity. A easureent of this velocity can be ade by several ethods, one of which is described in Section C. When the gun is rigidly held (Section D), this uzzle velocity is the speed of the dart. But the dart can also be released fro the gun by the slow dissolving of an aspirin tablet. So the gun can be fired fro the suspended launcher, which is allowed to recoil. When fired this way, the speed of the dart is not the uzzle velocity, because the launcher will be oving in the opposite direction. (See Section E) Pendulu Suspended auncher Dart Gun Fig. 1 Section B -Measuring the velocity of the block To get the initial recoil velocity V of pendulu plus bullet, we use the law of conservation of oentu. The vectors of interest are one diensional (the horizontal) so we siply designate the as signed nubers. All otion vectors directed fro the gun toward the block will be positive. Denoting the ass and velocity of the bullet with sall letters, the ass and velocity of the block with capital letters, we have v = ( + M)V Eq. (1) We cannot use conservation of kinetic energy when the dart strikes the block. It is left to you to explain why not (see Questions). However, the dart-block syste, after the inelastic collision has occurred, has an initial kinetic energy: 13103

3 Physics Departent Handout -10 T = ( + M)V / Eq. () Upon ipact the pendulu swings back and up without rotating, because of the four suspending cords. Eventually its center of ass coes instantaneously to rest, having risen a vertical distance h. Kinetic energy existing just after the collision has been converted into potential energy: U = ( + M)gh Eq. (3) The suspending cords have a length of several feet, while the pendulu rises only a sall distance h. It is therefore considerably ore accurate to easure the horizontal distance x that the block oves and infer by siple geoetry the vertical height h (see Fig ). Fro this geoetry we have that x ( l h) Eq. (4) h x Fig. A siple ethod allows x to be easured fairly accurately. A light glider slides on the eter stick at the base of the apparatus. The glider can be placed, after soe trial, so that the pendulu just nudges it along at the end of the pendulu swing. This leaves the glider arking the horizontal excursion of the pendulu, while the effect of dragging the glider is reduced to a negligible iniu. With this ethod of easuring x, we have a way of easuring h for any inelastic collision of the dart with the pendulu. Because the length of the vertical distance is so uch larger than h, the exact Eq. (4) is easily shown to be approxiated by the expression

4 Physics Departent Handout -10 h = x / Eq. (5) Once h is obtained fro Eq. (5), it can be used in Eq. (3), and because this is equal to the kinetic energy iparted by the dart, it applies for any collision in which the dart sticks to the block. Section C - Measureent of the Muzzle Velocity. The uzzle velocity is the speed of the dart relative to the dart gun when it is fired. It is easured as described below with the dart gun rigidly held. Put this relative velocity will be the sae whether the gun is fixed or allowed to recoil. So it is a constant of the apparatus that you want to easure for your particular dart gun soewhat carefully. To do so, the dart gun is ounted on a test stand separate fro the pendulu apparatus so that it fires vertically downward. The trajectory of the dart crosses two light beas that operate a counter. You will recognize this as a technique that was used in an earlier free fall experient. The two light beas, each with a sensing photocell, are separated by a vertical distance of about 60 c (you easure the exact distance). The counter is cleared to zero once the dart is placed in the dart gun, then the dart is fired. The counter will record the nuber of illiseconds it takes for the dart to cross the two beas, and fro this, after the correction for gravity; you obtain the speed of the dart. The average speed needs to be corrected for the gravitational acceleration during the flight, which adds directly to the speed because of the vertical trajectory. To ake this correction, recall that the dart will get an increent in velocity equal to gt (where g = 980c/s ) in the tie t that it takes to cross the beas. If the velocity of the dart is the uzzle velocity V 0 as it passes the first photocell, and v 0 + gt as it passes the second, the average velocity is v 0 + gt/. Equate this to the distances divided by the tie, giving and we can write this as v gt / d / t Eq. (6) 0 d d gt v 0 gt / 1 Eq. (7) t t d The second for of Eq.(7) has been written to show what the size of the correction looks like (if g were zero, the expression in parentheses would be unity). For the exaple value of d = 60c, t will be about 0.l sec. Calculate the correction (roughly) and convince yourself that it will be of the order of ten percent or less. This cannot be neglected, but also indicates that the approxiation that the velocity is only v 0 at the first photocell (when the dart has already begun accelerating for a few centieters) is reasonable. So Eq. (7) will be used to easure the uzzle velocity, v

5 Physics Departent Handout -10 Section D - Experient with Recoilless Firing. When the dart gun is held rigidly so it cannot recoil, the uzzle velocity v 0 found in Section C is equal to the velocity v of the dart ipinging on the block. But we can ake an independent easureent of v using the ethod of Section B. When V is obtained by equating the easured potential energy, Eq. (3) to the kinetic energy, Eq. (), then v can be written by Eq. (1) as ( M ) V M v (1 ) V Eq. (8) Therefore, that easureent of V can be used to calculate v, copare it to the uzzle velocity v 0. All the approxiations of Section B are ade better as the ass of the target is increased. So it is recoended that additional weight of about 00g be hung on the target pendulu. Then the easureents that need to be ade are: (1) Weigh the total target assebly (M). () Weigh the ass of the dart (). (3) Measure the vertical distance fro the top frae to the center of the pendulu target. (4) Fire the dart at the target fro the gun held rigidly, in your hand, to prevent recoil. (5) Fro the distance X that the target oves, calculate the velocity v of the dart and copare it to the uzzle velocity v 0. Section E Experient with Recoil. The easureents ade in this experient will be siilar to those of Section D. However, the dart gun here is suspended just like the target pendulu. The trigger is released by the slow dissolution of a wetted aspirin tablet, so that the gun dart syste is isolated with respect to horizontal otion at the instant of firing. This iparts a different velocity to the dart than the uzzle velocity. When the dart is fired, it oves to the left with velocity v. The launches and gun recoil to the right with velocity v. Because oentu is conserved, and was originally zero, in the horizontal direction, v v The kinetic energy after the release coes fro the unknown stored potential energy of the copressed spring before the release. Rather than deal with the energy equation, we can instead look at the release of the dart in the oving syste of the launcher. In that syste the dart has the uzzle velocity, v 0. Now the launcher syste is oving away fro the target with velocity

6 Physics Departent Handout -10 v v Eq. (9) as seen in the laboratory. To get the laboratory velocity of anything whose velocity is known in the launcher syste, we siply subtract (i.e., add vectorially) v as given by Eq. (9). In particular, the laboratory velocity of the dart is Therefore or v v0 v v0 v Eq.(l0) v( 1 ) v v0 v Eq. (11) (1 / ) 0 Notice that v will be less than the velocity found in Section D because of the recoil. If we ake very large, in other words look at the recoilless experient as one where the entire earth recoiled, we get v v 0, the expression of Section D. The procedure for this experient is the sae as in Section D, so you don t need to weigh anything except the launcher with the dart gun together, without the dart in it. This is the ass. Then suspend the launcher and load the dart. ock the trigger by inserting an aspirin. When the hanging objects have quieted down, use the edicine dropper to wet the aspirin, and wait for the dart to release. Fro the distance X that the target and dart ove this tie, you get the velocity v. This tie, copare v to the expression given in Eq.11. Procedure: Preliinaries. Measure (it should be the sae for both pendula) And the individual asses of pendulu carts, dart, and gun. (Soe of these ay be given). Measure also the distance through which the gun spring is copressed when the dart is cocked. This can easily be done by inserting the dart in the gun, standing the whole on the suction cup, and easuring the overall height, both with the dart un-cocked (resting loosely in the gun) and cocked. (1) Clap gun rigidly, fire dart onto pendulu cart and easure recoil distance. Copute dart s initial speed by the ethod outlined. () Mount the gun in the second pendulu suspension, fire dart by the hands off ethod and easure recoil distances of both pendulus. Fro the two recoils, copute the dart s speed on leaving the launch pendulu and on striking the target pendulu, respectively. (3) Reverse pendulu cart in the strings so that the slanted face of the target plate faces the gun. Clap gun rigidly at the angle necessary to fire it perpendicular to the slanted face. Fire (anually) and easure pendulu recoil. Calculate effective dart input speed in this case. (This is the

7 Physics Departent Handout -10 horizontal coponent of the dart s velocity. Analysis: Part (1) gives the uzzle velocity of the launcher. Calculate the average force, which the launching spring exerts on the dart, as follows. The deterination of the speed given to the dart tells you its kinetic energy, which is the work done by the spring in launching it. You have easured the distance through which the launching force acts; the work done in launching is that distance ties the average force applied. In part (), copare the values of dart speed (on leaving gun and on arriving at target) with each other, and with the value deterined in part (l). In part (3), copare deterination of effective dart speed with the horizontal coponent of the launch velocity as found in part (1). In each part (1, and 3) calculate the total kinetic energy of the whole syste both before and after the collision. Calculate the aount of kinetic energy created or destroyed in the collision. (Note that part involves two collisions... that between dart and launcher, that between dart and target.) In each case, account for the change in total kinetic energy if E is destroyed, where did it go? If created, where did it coe fro?