A heavy wooden block rests on a flat table and a high-speed bullet is fired horizontally into the block, the bullet stopping in it. How far will the block slide before coming to a stop? The mass of the bullet is 10.5 g, the mass of the block is 10.5 kg the bullet’s impact speed is 750 m/s, and the coefficient of kinetic friction between the block and the table is 0.220. (Assume that the bullet does not cause the block to spin.)

find the intial momentum of the block/bullet after impact, from that the velocity of the block. YOu now know that

KEblockAndBullet=mu*totalmass*g*distance
solve for distance.

Once sliding begins, there is a friction force equal to

f = (M+m)*g*0.22 = 22.7 N

The initial speed V' of the block-with-bullet, after impact is given by the law of conservation of momentum:
m*750 = (M+m)V
V' = (.0105/10.51)*750 = 0.75 m/s

As a last step, equate the kinetic energy of the sliding block-plus-bullet to the work done against friction.

(1/2)(M+m)V'^2 = f X

and calculate the sliding distance X.

m and M are bullet and block mass, respectively.

1/2 (10.5 kg) (750 m/s) + 1/2 (.0105 kg) (750 m/s) = .220 N * 10.5105 kg (9.81) distance

is this what i'm solving for?

so the distance would be...... 2.96 m ?

No. You use the velocity V' AFTER collision (0.75 m/s), and you square it, as in the formula I wrote.

Your friction force f also looks wrong

To find out how far the block will slide before coming to a stop, we need to consider the forces acting on the block and calculate the frictional force opposing its motion.

First, let's calculate the force exerted by the bullet on the block. The force is equal to the change in momentum, which is the mass of the bullet multiplied by its initial speed:

Force = Mass * Change in velocity
Force = 10.5 g * 750 m/s

To convert the mass of the bullet to kilograms (kg):
1 kg = 1000 g
10.5 g = 10.5 / 1000 kg

Now we can calculate the force exerted by the bullet on the block:
Force = 10.5 / 1000 kg * 750 m/s

Next, we need to calculate the frictional force opposing the motion of the block. The frictional force can be found using the formula:

Frictional Force = Coefficient of Kinetic Friction * Normal Force

The normal force is the force exerted by the table on the block, which is equal to the weight of the block. So:

Frictional Force = Coefficient of Kinetic Friction * Weight of the block

Now let's calculate the weight of the block:
Weight = Mass * Acceleration due to gravity
Weight = 10.5 kg * 9.8 m/s²

Lastly, we can calculate the distance traveled by the block before coming to a stop. The work done by the frictional force is equal to the product of the magnitude of the frictional force and the distance traveled:

Work = Frictional Force * Distance

Since work is also equal to the force applied multiplied by the distance traveled, we can equate the two:

Force * Distance = Frictional Force * Distance

Now we can solve for the distance:

Distance = (Force * Distance) / Frictional Force

Substituting the values we calculated earlier, we can find the distance traveled by the block before stopping.