A bullet of mass m=10 g is fired with initial speed vi = 400 m/s into a block
of mass M=2 kg initially at rest at the edge of a frictionless table of height h=1 m as
shown in the figure. The bullet remains in the block, and after the impact the block lands
a distance d from the bottom of the table. Determine the distance d.
To determine the distance d, we can start by finding the initial kinetic energy of the bullet and the final kinetic energy of the block-bullet system.
1. Find the initial kinetic energy of the bullet:
The initial kinetic energy (K1) of an object is given by the formula:
K1 = 1/2 * m * vi^2
Substituting the given values:
m = 10 g = 0.01 kg (converted from grams to kilograms)
vi = 400 m/s
K1 = 1/2 * 0.01 kg * (400 m/s)^2
K1 = 8 J
2. Find the final kinetic energy of the block-bullet system:
After the bullet is embedded in the block, the combined system has a new mass (M + m).
The final kinetic energy (K2) of an object is given by the formula:
K2 = 1/2 * (M + m) * vf^2
Since the block and bullet land together, the final velocity (vf) of the block-bullet system is the same as the velocity of the block when it lands.
3. Find the velocity of the block when it lands:
Using the principle of conservation of momentum, we can determine the velocity (v) of the block-bullet system just before it lands.
The initial momentum of the bullet (p1) is equal to the final momentum of the block-bullet system (p2).
p1 = p2
The momentum (p) of an object is given by the formula:
p = m * v
For the bullet:
p1 = m * vi
For the block-bullet system:
p2 = (M + m) * v
Equating p1 and p2:
m * vi = (M + m) * v
Solving for v:
v = (m * vi) / (M + m)
Substituting the given values:
m = 10 g = 0.01 kg
M = 2 kg
vi = 400 m/s
v = (0.01 kg * 400 m/s) / (2 kg + 0.01 kg)
v = 0.1996 m/s
4. Find the final kinetic energy of the block-bullet system (K2):
Using the calculated velocity (v), we can find the final kinetic energy (K2) of the block-bullet system.
K2 = 1/2 * (M + m) * v^2
K2 = 1/2 * (2 kg + 0.01 kg) * (0.1996 m/s)^2
K2 = 0.03992 J
5. Determine the distance d:
The change in potential energy (ΔPE) of the block-bullet system can be calculated using the formula:
ΔPE = m * g * h
Where:
g = acceleration due to gravity = 9.8 m/s^2
h = height = 1 m
ΔPE = 0.01 kg * 9.8 m/s^2 * 1 m
ΔPE = 0.098 J
The total mechanical energy (E) of a system is conserved when no external forces are acting. Therefore, the initial kinetic energy (K1) of the bullet is equal to the sum of the final kinetic energy (K2) of the block-bullet system and the change in potential energy (ΔPE).
K1 = K2 + ΔPE
8 J = 0.03992 J + 0.098 J
Now, we can solve for the distance d using the equation:
ΔPE = m * g * d
Substituting the values:
0.098 J = (2 kg + 0.01 kg) * 9.8 m/s^2 * d
Simplifying:
0.098 J = 2.01 kg * 9.8 m/s^2 * d
d = 0.098 J / (2.01 kg * 9.8 m/s^2)
d = 0.00495 m
Therefore, the distance d that the block lands from the bottom of the table is approximately 0.00495 meters.