Question

A dart of mass m is fired at and sticks into a block of mass M that is initially at rest on a rough, horizontal surface. The coefficient of kinetic friction between the block and the surface is μk. After the collision, the dart and the block slide a distance D before coming to rest. If the dart were fired horizontally, what would its speed be immediately before impact with the block? (Use any variable or symbol stated above along with the following as necessary: g.)

Answer 1

I will try to find a solution. First, we should apply the conservation of linear momentum

________________________________(1)

where

is the linear momentum before the collision

is the linear momentum of the system /is the block´s mass) after the collision. We now should apply (1)

_______________________(2)

________________________________________________________________

After the collision the dart and the block slide a distance D, then the work done by the friction force

must be equal to the change in Kinetic energy (theorem of work and energy)

_________________________________(3)

the work done by the friction force is given by

the final kinetic energy is zero and the initial kinetic energy ( is the velocity after the collision of the dart with the block) is given by,

Now, we must satisfy (3), hence

,

the velocity of the block and the dart is given by,

____________________________(4)

Now, substituting /4) in (2), we get the initial velocity of the dart (before the collision)

_______________________________________________________________________