An experiment is done to compare the initial speed of projectiles fired from high-performance catapults. the catapults are placed very close to a pendulum bob of mass 10.0 kg attached to a massless rod of length l. assume that catapult 1 fires a projectile of mass 5.50 g at speed v1 and catapult 2 fires a projectile of mass 10.0 g at speed v2. if the projectile from catapult 1 causes the pendulum to swing to a maximum angular displacement of 6.80 ∘ and the projectile from catapult 2 causes a displacement of 11.4 ∘, find the ratio of the initial speeds, v1v2 .

The conservation of momentum and energy allows finding the result for the relationship of the velocities of the balls fired by the two catapults is:

Given parameters

To find

The momentum is defined by the product of the massby the speed of the particle, without the system is isolated the momentum is conserved.

      P = mv

Where bold letters indicate vectors, p is momentum, m is mass, and v is velocity.

Let's solve the problem for each ball separately, let's start with case 1 the ball of m₁ = 5.50 10⁻³ kg.

The system is shaped by the ball and the pendulum, therefore it is an isolated system and the momentum is conserved.

Initial instant. Before crash

      p₀ = m₁ v₁

Final moment. Right after the crash

      = (m₁ + M) v

The momentum is preserved

      p₀ =

      m₁ v₁ = (m₁ + M) v

      v₁ =           (1)

Now we can use the conservation of energy for the pendulum with the embedded ball m1, see attached.  

Starting point. Just when they're coming out at the bottom

        Em₀ = K = ½ (m₁ + M) v²

Final point. The highest part of the movement

        = U = (m₁ + M) gh

Energy is conserved.

       Em₀ =

       ½ (m₁ + M) v² = (m₁ + M) gh

       v =

Let's use trigonometry to find the height of the pendulum, see attached

       h = L - L cos θ₁

Let's substitute

      v =  

Let's substitute in equation 1

We solve the case 2 ball of mass m₂ = 10 10⁻³ kg.

The system has the same approach: the conservation of the moment and then the conservation of energy, therefore the expression for the speed of this ball is:

They ask us for the relationship between the two speeds.

Let's calculate each velocity and then make the relationship.

Suppose a pendulum length value of L = 1 m, to make an explicit calculation.

       v₁ = 675.49 m / s

       v₂ = 622.46 m/s

The ratio of the speeds is:

         = 1,085

In conclusion, using the conservation of momentum and energy, we can find the relationship of the velocities of the balls fired by the two catapults is:

           = 1,085

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1.084

Explanation:

Given

mass of Pendulum M=10 kg

mass of bullet m=5.5 gm

velocity of bullet u

After collision let say velocity is v

conserving momentum we get

Conserving Energy for Pendulum

Kinetic Energy=Potential Energy

here from diagram

therefore

initial velocity in terms of v

For first case

for second case

Therefore

i.e.

0.3x0.21x0.08kg=mass 1 sheet

mass 500 = 500xmass 1

volume pile = 0.3x0.2x0.048

density=mass/volume= 500xmass 1/0.3x0.2x0.048

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