The Ballistic Pendulum

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  • Topic: Pendulum, Mass, Ballistics
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The Ballistic Pendulum|
Determining the initial speed of a projectile|
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THE BALLISTIC PENDULUM

ABSTRACT
The experiment was carried out to determine the initial speed of a projectile: i) by means of a ballistic pendulum.
ii) by measurements of the range and vertical distance of fall during its flight. The initial speed for the ballistic pendulum was found to be 5.0551±0.0008m/s and the initial speed for the pendulum was found to be 4.72±0.02 m/s.

INTRODUCTION:
i) Ballistic pendulum
A ball of mass m and velocity v₀ nets a pendulum bob of mass M. The total mass (M + m) acquires a velocity v₀ just after impact, and subsequently rises by a height h. Using the law of conversion of momentum, mv₀ = (M + m)v velocity v₀ (when neglecting friction) is given by: v₀ = (M+mm)2gl (1-cosθ

θ
θ

l - h
l - h

l
l

h
h

12mv2= mgh (by law of conversion of energy)
mv2= 2mghcosθ= adjacenthypotenuse
v2= 2ghlcosθ=l-hl*l
v=±2ghlcosθ=l-h
l-lcosθ=h
l1-cosθ=h
* h=l1-cosθ
* v=±2gh
= ±2gl(1-cosθ) ………..(1)

mv₀ = (M + m)v ……….(2)
Subtracting 1 into 2 we get:
mv₀ = (M + m) ± 2gl1-cosθ
v₀ = (M+mm) ± 2gl (1-cosθ

ii) Projectile
The projectile with initial horizontal velocity v₀ is projected in such a way that its vertical distance of fall is H and the range is R. The trajectory is illustrated in the diagram below. V₀
V₀
H
H
R
R

Using the kinematic equation s = ut + ½ at2 we get x = v₀t along the horizontal direction. sina-u= v₀then: x=v₀(t)+12(0)t2
a=0 x=v₀t+02
x=v₀t
and along vertical direction:
y=12gt2u=g=0
When t is time of flight, then x = 2 and y = 1
Giving v₀ = lg2h1=12gt2; t = 2Hg
v₀ = xt=l2Hg=l2Hg12
=lg2h

IMAGE OF A BALLISTIC PENDULUM

METHODS AND MATERIALS:
MATERIALS
Assembled ballistic pendulum, steel ball, ramrod, metre rule, electronic balance, tape measure, 2 plain papers, carbon paper and Sellotape. METHODS
i) The Ballistic pendulum (Dynamic method)
It was ensured that the ballistic pendulum apparatus was on a level surface by making sure the plumb bob was at zero degrees. Mass M of the Ballistic pendulum and the length R from the pivot to the centre of the pendulum cage were written on the ballistic pendulum base. M and R were noted and mass m of the steel ball was measured. The pendulum was swung out of the way and the steel ball placed in the piston. With the ball in the piston, the ramrod was used to load the launcher to long range. The pendulum was swung back and the angle indicator for the pendulum swing reset to the initial position. The pendulum was firmly held at the base to the bench and the launcher was fired by pulling the string attached to the trigger upwards. The angle θ was noted from the angle indicator. The ball was released from the pendulum and then the procedure was repeated nine times to obtain values of θ. The average value of θ was obtained, denoted by θ and hence standard deviation of θ was calculated.

ii) The projectile (kinematic method)
The pendulum was swung out of the way and the steel ball placed in the piston. With the ball in the piston, the ramrod was used to load the launcher to long range. The trigger was pulled and the ball fired to find its approximate range. The position where the ball hits the floor was noted and carbon paper was taped between two papers (carbon side down) to the floor such that the point where the ball hit the floor was approximately at the centre. Ten values of R were recorded and the average calculated also. The vertical distance of fall, H, was also recorded. R and H were measured from the part of the ball that hit the ground.

RESULTS:
i) The ballistic pendulum
Mass of pendulum, M = 245.05 ± 0.01g,0.24505 ± 0.00001 kg Pendulum length, l = 31.7 ± 0.1 cm,0.317 ± 0.001 m
Mass of steel ball, m = 65.90 ± 0.01g,0.0659 ± 0.00001 kg

θi (°) ±0.5|...
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