A balistic pendulum is made from a block of wood with a mass of 2.78 kg. A bullet with a mass of 39 g is shot into the block causing it to rise 29.1 cm. What was the speed of the bullet?
find the speed of the pendulum at the bottom
mgh=1/2 m vi^2
now, knowing vi, from conservation
(Mb+mblock)vi=Mb*V
solve for the speed of the bullet V
momentum before = momentum just after
Vb bullet
v block and bullet
.039 Vb = (2.78+.039)v = 2.82 v
now work on potential and kinetic energy after
kinetic just after = .5 * 2.82 v^2
potential when stopped at top = m g h =2.82 * 9.81 * .291 = 8.05 Joules
so
1.41 v^2 = 8.05
v = 2.39 m/s
now back to that initial momentum equation
.039 Vb = 2.82 (2.39)
Vb = 173 m/s
To find the speed of the bullet, we can use the concept of conservation of momentum. The momentum of an object is the product of its mass and its velocity.
In this case, we have a bullet with a mass of 39 g (or 0.039 kg) and a wooden block with a mass of 2.78 kg. Before the collision, the bullet is moving with some initial velocity, and after the collision, the bullet becomes embedded in the block, causing it to rise.
Let's denote the initial velocity of the bullet as v (which is what we want to find) and the final velocity of the bullet-block system as V. According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
Before the collision:
Momentum of bullet = mass of bullet × velocity of bullet
After the collision:
Momentum of bullet + Momentum of block = (mass of bullet + mass of block) × velocity of the bullet-block system
Using the conservation of momentum, we can write the equation as:
(mass of bullet) × (initial velocity of bullet) = (mass of bullet + mass of block) × (final velocity of bullet-block system)
Simplifying the equation, we have:
0.039 kg × v = (0.039 kg + 2.78 kg) × V
0.039 kg × v = 2.819 kg × V
Now, we can solve for V by dividing both sides of the equation by 2.819 kg:
V = (0.039 kg × v) / 2.819 kg
V = 0.0139 v
Next, we can use the fact that the block rises 29.1 cm (or 0.291 m) to find the velocity V. When an object is shot into a block, the increase in potential energy of the block is equal to the initial kinetic energy of the bullet. Therefore, we can use the equation:
Change in potential energy of the block = initial kinetic energy of the bullet
The change in potential energy is given by:
(mass of block × acceleration due to gravity × change in height)
Substituting the values, we have:
(2.78 kg × 9.8 m/s^2 × 0.291 m) = 0.5 × (0.039 kg) × (V^2)
Simplifying the equation, we find:
V^2 = ((2.78 kg × 9.8 m/s^2 × 0.291 m) / (0.039 kg × 0.5))
Finally, we can solve for V by taking the square root of both sides of the equation:
V = √(((2.78 kg × 9.8 m/s^2 × 0.291 m) / (0.039 kg × 0.5)))
Calculating this expression, we find V = 1.77 m/s.
Therefore, the speed of the bullet is approximately 1.77 m/s.