Question

You throw a 20.0 g dart horizontally into the side of a small cardboard box of mass 45.0 g that is initially
at rest on a tabletop. The coefficient of the kinetic friction between the cardboard box and the tabletop
is 0.35. The dart remains stuck in the box, which slides 0.530 m on the tabletop before stopping. How
fast did you throw the dart?

Answer 1

Mass of dart, Md = 20 g
Mass of box, Mb = 45 g
The initial velocity of the dart = u
The final velocity of the dart + box = v
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Mass of the dart+box system, M = Md + Mb
Frictional force on the dart+box system, Ff = k * (Md + Mb) * g
Deceleration of the dart+box system, a = - Ff/M
= - [k * (Md + Mb) * g] / (Md + Mb)
= - k * g
= - 0.35 * 9.81
= - 3.43 m/s²
The initial velocity of the dart+box system = v
The final velocity of the dart+box system, vf = 0
Distance travelled, s = 0.530 m

Using the formula, vf² - v² = 2 * a * s,
0² - v² = 2 * -3.43 * 0.53
v² = 3.64
v = SQRT[3.64]
= 1.91 m/s
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Initial momentum of the system, Pi = Md * u
Final momentum of the system, Pf = (Md + Mb) * v
Using the conservation of momentum, Pi = Pf
Md * u = (Md + Mb)
20 * u = (20 + 45) * 1.91
u = 124 / 20
= 6.20 m/s