A ballistic pendulum is used to measure the speed of bullets. It comprises a heavy block of wood of mass M suspended by two long cords. A bullet of mass m is fired into the block in the horizontal direction. The block, with the bullet embedded in it, swings upward (see figure below). The center of mass of the combination rises through a vertical distance h before coming to rest momentarily. In a particular experiment, a bullet of mass 48.0 g is fired into a wooden block of mass 11.2 kg. The block–bullet combination is observed to rise to a maximum height of 19.6 cm above the block's initial height.
(a) What is the initial speed of the bullet?
m/s
(b) What is the fraction of initial kinetic energy lost after the bullet is embedded in the block? (Enter your answer to at least three significant figures.)
(KEm-KEM+m)/KEm=
(a) How did the bullet get 48.0 g of mass? Did it go on a cheeseburger binge before being fired? But more importantly, to find the initial speed of the bullet, we can use the principle of conservation of momentum. Since momentum is conserved, we have:
initial momentum of bullet = final momentum of block-bullet combination
So, we can write:
(mass of bullet) * (initial speed of bullet) = (mass of block-bullet combination) * (final velocity of block-bullet combination)
Since the final velocity is zero, we can simplify it to:
(initial speed of bullet) = (mass of block-bullet combination) * (final velocity of block-bullet combination) / (mass of bullet)
Remember to convert the mass of the bullet to kilograms!
(b) Ah, the fraction of initial kinetic energy lost. Well, if energy were easy to lose, I would have misplaced my comedy skills ages ago! To find the fraction of initial kinetic energy lost, we can use the equation:
fraction of initial kinetic energy lost = (initial kinetic energy - final kinetic energy) / initial kinetic energy
But first, we need to calculate the initial kinetic energy using the formula:
initial kinetic energy = (1/2) * (mass of bullet) * (initial speed of bullet)^2
And the final kinetic energy is simply zero, considering the block-bullet combination has come to rest.
To find the initial speed of the bullet, we can use the principle of conservation of momentum.
The initial momentum of the bullet-block system is equal to the final momentum, after the bullet is embedded in the block and they move together as a single unit.
Let's denote the initial speed of the bullet as v and the final combined velocity of the bullet-block system as V.
The principle of conservation of momentum gives us:
(mass of bullet) * (initial velocity of bullet) = (mass of bullet + mass of block) * (final velocity of block-bullet system)
m * v = (m + M) * V
Plugging in the given values:
0.048 kg * v = (0.048 kg + 11.2 kg) * V
0.048v = 11.248V
Now, let's consider the conservation of mechanical energy to determine the value of V.
The mechanical energy of the system is conserved, which means the initial kinetic energy of the bullet and the initial potential energy due to the block's initial height must equal the final kinetic energy of the combined block-bullet system and the final potential energy at the maximum height.
Initial kinetic energy + Initial potential energy = Final kinetic energy + Final potential energy
(0.5 * mass of bullet * (initial velocity of bullet)^2) + (mass of block * g * h) = 0.5 * (mass of bullet + mass of block) * (final velocity of block-bullet system)^2 + 0
(0.5 * 0.048 kg * v^2) + (11.2 kg * 9.8 m/s^2 * 0) = 0.5 * (0.048 kg + 11.2 kg) * V^2
0.024v^2 = 5.624V^2
Dividing both sides of the equation by 5.624:
0.00427v^2 = V^2
Taking the square root of both sides:
0.0654v = V
Now we have two equations involving v and V:
0.048v = 11.248V (equation based on momentum conservation)
0.0654v = V (equation based on energy conservation)
We can solve these equations simultaneously to find the initial speed of the bullet (v) and the final combined velocity (V).
Let's substitute the value of V from the second equation into the first equation:
0.048v = 11.248 * 0.0654v
0.048v = 0.7340992v
Dividing both sides by v:
0.048 = 0.7340992
Now we can solve for v:
v = 0.7340992 / 0.048
v ≈ 15.292 m/s
(a) The initial speed of the bullet is approximately 15.292 m/s.
Now, let's find the fraction of initial kinetic energy lost after the bullet is embedded in the block.
The initial kinetic energy (KEm) of the bullet is given by:
KEm = 0.5 * m * (initial velocity of bullet)^2
Plugging in the values:
KEm = 0.5 * 0.048 kg * (15.292 m/s)^2
KEm ≈ 5.872 J
The final kinetic energy (KEm) of the combined block-bullet system is given by:
KEm = 0.5 * (mass of bullet + mass of block) * (final velocity of block-bullet system)^2
Plugging in the values:
KEm = 0.5 * (0.048 kg + 11.2 kg) * (15.292 m/s)^2
KEm ≈ 136.618 J
The fraction of initial kinetic energy lost is given by:
(KEm - KEM) / KEm
Substituting the values:
(136.618 J - 5.872 J) / 136.618 J
= 130.746 J / 136.618 J
≈ 0.957
(b) The fraction of initial kinetic energy lost after the bullet is embedded in the block is approximately 0.957.
To solve this problem, we can use the principle of conservation of momentum and the principle of conservation of mechanical energy.
1. Let's start by calculating the initial speed of the bullet.
- The equation for conservation of momentum is:
(m_bullet * v_bullet) = (M_block + m_bullet) * V_combined
where:
m_bullet = mass of the bullet = 48.0 g = 0.048 kg
v_bullet = initial velocity of the bullet (which we need to find)
M_block = mass of the wooden block = 11.2 kg
V_combined = velocity of the block-bullet combination after the collision
- Next, we need to find the velocity of the block-bullet combination, V_combined. We can use the principle of conservation of mechanical energy.
- The equation for conservation of mechanical energy is:
(KEm_bullet + KEm_block) = (PE_block-bullet_combination)
where:
KEm_bullet = kinetic energy of the bullet before the collision
KEm_block = kinetic energy of the block before the collision
PE_block-bullet_combination = potential energy of the block-bullet combination at maximum height
- The kinetic energy of an object is given by:
KEm = (1/2) * mass * (velocity)^2
- The potential energy of an object is given by:
PE = mass * gravity * height
where:
mass = total mass of the block-bullet combination
gravity = acceleration due to gravity (approximately 9.8 m/s^2)
height = maximum height reached by the block-bullet combination (19.6 cm = 0.196 m in this case)
Now, let's solve for the unknowns:
(a) What is the initial speed of the bullet?
- From the conservation of momentum equation:
(0.048 kg * v_bullet) = (11.2 kg + 0.048 kg) * V_combined
- Rearranging the equation:
v_bullet = [(11.2 kg + 0.048 kg) * V_combined] / 0.048 kg
- Remember, we still need to find V_combined.
(b) What is the fraction of initial kinetic energy lost after the bullet is embedded in the block?
- First, let's find the initial kinetic energy of the bullet using the formula:
KEm_bullet = (1/2) * m_bullet * (v_bullet)^2
- Next, let's find the initial kinetic energy of the block using the formula:
KEm_block = (1/2) * M_block * (V_combined)^2
- Finally, let's find the potential energy of the block-bullet combination at maximum height using the formula:
PE_block-bullet_combination = (M_block + m_bullet) * gravity * height
- The fraction of initial kinetic energy lost is given by:
(KEm_initial - KEm_final) / KEm_initial
where KEm_initial is the initial kinetic energy and KEm_final is the final kinetic energy (which is equal to zero at maximum height).
Now, you can substitute the known values into the equations and calculate the answers for (a) and (b) using these steps.
(a) momentum is conserved ... .0480 Vib = (11.2 + .048) Vbb
... (Vbb)^2 = 2 * g * 0.0196
(b) KEb = 1/2 * 0.0480 * (Vib)^2 ... this is KEm
... KEbb = 11.248 * g * 0.0196 ... this is KEM+m