An 8.00g bullet is fired at 220m/s into a 250g wooden block that is initially at rest. The bullet remains in the block and after the collision the two slide up a 30° incline.
a. Solve for the velocity of the bullet-block after the collision. (ans: 6.82m/s) b. Determine distance along the incline the bullet-block travel if the incline is
frictionless. (ans: 4.74m)
I solved for part a, and got 6.82 m/s just fine. But I can't seem to solve for b. I think this is a kinematic since without friction it isn't necessary to make a free body diagram. I multiplied 6.82cos30 and 6.82sin30 to get the respective initial velocities in the x and y direction. There seems to be no acceleration in the x since the block was struck at rest so the final velocity will be the same..... But you still don't have enough to find the change in x or y. What am I missing?
-Mollie
To solve for part b, you are correct in using kinematics since there is no friction involved. However, you need to consider the conservation of momentum and energy equations to find the necessary information.
Here's a step-by-step guide on how to solve for the distance along the incline that the bullet-block travel:
1. Find the initial momentum of the system before the collision.
- The initial momentum is given by the product of the bullet's mass and velocity.
- Initial momentum (P_initial) = bullet mass (m_bullet) * bullet velocity (v_bullet)
2. Use the conservation of momentum to find the final velocity of the bullet-block system after the collision.
- The momentum of the system after the collision should be equal to the initial momentum.
- Final momentum (P_final) = bullet-block mass (m_bullet-block) * final velocity (v_final)
- Set P_final equal to P_initial: m_bullet * v_bullet = (m_bullet-block) * v_final
- You already solved part a and found the final velocity (v_final = 6.82 m/s) for the bullet-block system.
3. Find the change in potential energy of the bullet-block system.
- The change in potential energy is equal to the work done by gravity when the system moves up the incline.
- Change in potential energy (ΔPE) = m_bullet-block * g * h
- The height (h) is not given directly, but you can use the angle of the incline (30°) to find it.
- h = length of incline * sin(angle)
- Note: The vertical component of the velocity is affected by gravity, so you need to use the vertical component of the final velocity.
4. Use the conservation of energy to relate the change in potential energy to the kinetic energy gained by the bullet-block system.
- The total energy before the collision is kinetic energy of the bullet only.
- The total energy after the collision is kinetic energy of the bullet-block system.
- Change in potential energy = kinetic energy of the bullet-block system
- ΔPE = (1/2) * m_bullet-block * (v_final)^2
5. Solve for the length of the incline:
- Set the equations from steps 3 and 4 equal to each other:
- m_bullet-block * g * h = (1/2) * m_bullet-block * (v_final)^2
- Divide both sides by m_bullet-block and rearrange the equation:
- g * h = (1/2) * (v_final)^2
- Since h = length of incline * sin(angle), substitute it in the equation:
- g * length of incline * sin(angle) = (1/2) * (v_final)^2
- Now, you can solve for the length of the incline (length of incline = distance traveled).
6. Calculate the length of the incline to find the distance along the incline.
- Plug in the known values: g = acceleration due to gravity (9.8 m/s^2), angle = 30°, and v_final = 6.82 m/s.
- Solve for the length of the incline.
Following these steps should allow you to solve for the distance along the incline that the bullet-block travel (part b).