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10 February, 20:37

A ballistic pendulum is a device used to measure the speed of a projectile, such as a bullet. The projectile, of mass m, is fired into a large block (of wood or other material) of mass M, which is suspended like a pendulum. (Usually M is somewhat greater than m). As a result of the collision, the pendulum and projectile together swing up to a maximum height h. Determine the relationship between the initial horizontal speed of the projectile, v and the maximum height h.

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  1. 10 February, 20:41
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    v = ((m + M) / m) * √ (2*g*h)

    Explanation:

    Given

    m = mass of the projectile

    M = mass of the ballistic pendulum

    v = initial speed of the projectile

    v' = speedof the system (pendulum + projectile) after the inelastic collision

    h = maximum height reached for the system

    Knowing that is an inelastic collision we have

    m*v + M * (0) = (m+M) * v'

    ⇒ v' = m*v / (m+M)

    After the collision, we apply the Principle of the Conservation of Energy

    Ki + Ui = Kf + Uf

    where

    Ui = Kf = 0 J

    then

    Ki = Uf

    0.5 * (m+M) * v'² = (m+M) * g*h

    ⇒ 0.5*v'² = g*h

    ⇒ v'² = 2*g*h

    ⇒ (m*v / (m+M)) ² = 2*g*h

    ⇒ v = ((m+M) / m) * √ (2*g*h)
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