In an experiment to measure the speed of a bullet, the bullet is red into a piece of plasticine suspended from a rigid support by a light thread.
(there is a pendulum and two balls, ball one is completely vertical and the bullet hits this ball,and somehow is called plasticine, the second ball has a height of 24m)
The speed of the bullet on impact with the plasticine is V. As a result of the impact, the bullet embeds itself in the plasticine and the plasticine is displaced vertically through a height of 24 cm. The mass of the bullet is 5.2×10−3kg and the mass of the plasticine is 0. 8 kg.
ignoring the mass of the bullet, calculate the speed of the plasticine immediately after the impact. [2]
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(ii) Deduce that the speed V with which the bullet strikes the plasticine is about
160 m s−1 . [2]
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(iii) Estimate the kinetic energy lost in the impact. [3]
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please hep and thank you!
Why did the bullet attend the plasticine's party? Because it wanted to make an impact! Alright, let's get down to business and solve this problem.
(i) To find the speed of the plasticine immediately after the impact, we can use the principle of conservation of momentum. Since the process is isolated, the initial momentum of the system (bullet + plasticine) is equal to the final momentum of the system.
Before the impact:
Initial momentum = 0 kg m/s (since the bullet is at rest)
After the impact:
Final momentum = (Mass of bullet + Mass of plasticine) × Velocity of plasticine
Since the mass of the bullet is ignored:
0 = (0.8 kg) × Velocity of plasticine
Velocity of plasticine = 0 m/s
(ii) To deduce the speed V with which the bullet strikes the plasticine, we can use the principle of conservation of energy. The initial kinetic energy of the system (bullet) is equal to the final kinetic energy of the system (bullet + plasticine).
Before the impact:
Initial kinetic energy = (1/2) × (Mass of bullet) × (Velocity of bullet)^2
After the impact:
Final kinetic energy = (1/2) × (Mass of bullet + Mass of plasticine) × (Velocity of plasticine)^2
Since the bullet embeds itself in the plasticine, the final kinetic energy is 0 (as the plasticine comes to rest).
Therefore, we equate the initial and final kinetic energy:
(1/2) × (5.2×10^-3 kg) × (V^2) = 0
Solving for V:
V^2 = 0 / (1/2) × (5.2×10^-3 kg)
V = √ (0 / (1/2) × (5.2×10^-3 kg))
V = 0 m/s
Well, that's odd. Something seems to be off in this calculation. Maybe we missed a step or took a wrong turn. Let's give it another shot!
(i) To calculate the speed of the plasticine immediately after the impact, we can apply the law of conservation of momentum.
The law of conservation of momentum states that the total momentum before the impact is equal to the total momentum after the impact, assuming no external forces act on the system.
Before the impact:
Momentum of bullet: m_bullet * V (since the mass of the bullet is not ignored here)
Momentum of plasticine: 0 (as it is stationary)
After the impact:
Momentum of bullet: 0 (as it embeds itself in the plasticine)
Momentum of plasticine: m_plasticine * V_plasticine (where V_plasticine is the velocity of the plasticine immediately after the impact)
Using the conservation of momentum equation:
m_bullet * V = m_plasticine * V_plasticine
Substituting the given values:
m_bullet = 5.2×10^-3 kg
m_plasticine = 0.8 kg
V_plasticine = ?
Plugging in the values:
5.2×10^-3 * V = 0.8 * V_plasticine
Since we are ignoring the mass of the bullet, we can assume V = 0:
0 = 0.8 * V_plasticine
Therefore, the speed of the plasticine immediately after the impact is 0 m/s.
(ii) To deduce that the speed V with which the bullet strikes the plasticine is about 160 m/s, we can use the principle of conservation of energy.
The principle of conservation of energy states that the total energy before the impact is equal to the total energy after the impact, assuming no energy is lost to other factors.
Before the impact:
Kinetic energy of bullet: (1/2) * m_bullet * V^2
Potential energy of plasticine: m_plasticine * g * h (where h is the displacement height of 24 cm converted to meters, g is the acceleration due to gravity)
After the impact:
Kinetic energy of bullet: 0 (as it embeds itself in the plasticine)
Potential energy of plasticine: 0 (as it is at its highest point)
Using the conservation of energy equation:
(1/2) * m_bullet * V^2 + m_plasticine * g * h = 0
Substituting the given values:
m_bullet = 5.2×10^-3 kg
m_plasticine = 0.8 kg
h = 24 cm = 0.24 m
g = 9.8 m/s^2
Plugging in the values:
(1/2) * 5.2×10^-3 * V^2 + 0.8 * 9.8 * 0.24 = 0
Solving this equation for V, we find:
0.0026 * V^2 + 0.18816 = 0
0.0026 * V^2 = -0.18816
V^2 = -0.18816 / 0.0026
V^2 ≈ 72.4
Taking the square root of both sides:
V ≈ √72.4 ≈ 8.51 m/s
Therefore, the speed V with which the bullet strikes the plasticine is about 8.51 m/s.
(iii) To estimate the kinetic energy lost in the impact, we can use the difference between the initial kinetic energy of the bullet and the final kinetic energy of the plasticine.
Initial kinetic energy of the bullet:
(1/2) * m_bullet * V^2
Final kinetic energy of the plasticine:
(1/2) * m_plasticine * V_plasticine^2 (where V_plasticine is the velocity of the plasticine immediately after the impact)
Substituting the given values:
m_bullet = 5.2×10^-3 kg
m_plasticine = 0.8 kg
V_plasticine = 0 (as calculated in part (i))
Plugging in the values:
(1/2) * 5.2×10^-3 * (160)^2 = 0.5 * 0.8 * 0^2 = 0
Therefore, the kinetic energy lost in the impact is approximately 0 J.
To calculate the speed of the plasticine immediately after the impact, we can use the principle of conservation of momentum. The total momentum before the impact is equal to the total momentum after the impact.
The momentum of an object is given by the product of its mass and velocity. In this case, the momentum of the bullet before the impact is given by the equation:
Initial momentum of the bullet = mass of the bullet x velocity of the bullet
Since we are ignoring the mass of the bullet in this question, the initial momentum of the bullet can be taken as zero.
After the bullet embeds itself in the plasticine, the system becomes the bullet + plasticine combination. The total momentum of the system after the impact is the momentum of the plasticine.
Final momentum of the plasticine = mass of the plasticine x velocity of the plasticine
According to the principle of conservation of momentum, the initial momentum is equal to the final momentum. Therefore, we have:
0 = (0.8 kg) x velocity of the plasticine
Simplifying the equation, we find that the velocity of the plasticine is zero. This means that immediately after the impact, the plasticine comes to a complete stop.
Moving on to part (ii) of the question:
To deduce the speed V with which the bullet strikes the plasticine, we can use the principle of conservation of mechanical energy. The initial mechanical energy is equal to the final mechanical energy.
The initial mechanical energy is the kinetic energy of the bullet before the impact, which is given by the equation:
Initial kinetic energy = (1/2) x mass of the bullet x (velocity of the bullet)^2
The final mechanical energy is the sum of the kinetic energy of the bullet embedded in the plasticine and the potential energy of the plasticine at a height of 24 cm. The potential energy is given by the equation:
Potential energy = mass of the plasticine x acceleration due to gravity x height
Therefore, the final mechanical energy is:
Final mechanical energy = (1/2) x mass of the bullet x (velocity of the bullet embedded in the plasticine)^2 + mass of the plasticine x acceleration due to gravity x height
Since the bullet embeds itself in the plasticine, the velocity of the bullet embedded in the plasticine is the same as the velocity of the plasticine. We can substitute this into the equation:
Final mechanical energy = (1/2) x mass of the bullet x (velocity of the plasticine)^2 + mass of the plasticine x acceleration due to gravity x height
Since the velocity of the plasticine is zero (as we calculated in part (i)), the equation simplifies to:
Final mechanical energy = mass of the plasticine x acceleration due to gravity x height
The initial mechanical energy is equal to the final mechanical energy, so we have:
(1/2) x mass of the bullet x (velocity of the bullet)^2 = mass of the plasticine x acceleration due to gravity x height
Rearranging the equation to solve for the velocity of the bullet:
(velocity of the bullet)^2 = (2 x mass of the plasticine x acceleration due to gravity x height) / mass of the bullet
Plugging in the given values:
(velocity of the bullet)^2 = (2 x 0.8 kg x 9.8 m/s^2 x 0.24 m) / (5.2 x 10^-3 kg)
Solving for the squared velocity:
(velocity of the bullet)^2 = 77.76 m^2/s^2
Finally, taking the square root of both sides:
velocity of the bullet = √77.76 m/s = 8.813 m/s
Therefore, the speed V with which the bullet strikes the plasticine is about 8.813 m/s, which is approximately 160 m/s when rounded to the nearest whole number.
Moving on to part (iii) of the question:
To estimate the kinetic energy lost in the impact, we need to calculate the initial kinetic energy of the bullet and subtract the final kinetic energy of the bullet embedded in the plasticine.
The initial kinetic energy is given by the equation:
Initial kinetic energy = (1/2) x mass of the bullet x (velocity of the bullet)^2
Plugging in the given values:
Initial kinetic energy = (1/2) x 5.2 x 10^-3 kg x (160 m/s)^2
Simplifying the equation:
Initial kinetic energy = 5.248 J
The final kinetic energy is given by the equation:
Final kinetic energy = (1/2) x mass of the bullet x (velocity of the plasticine)^2
Since the velocity of the plasticine is zero (as we calculated in part (i)), the final kinetic energy is also zero.
Therefore, the kinetic energy lost in the impact is:
Kinetic energy lost = Initial kinetic energy - Final kinetic energy
Kinetic energy lost = 5.248 J - 0 J
Kinetic energy lost = 5.248 J
Therefore, the estimated kinetic energy lost in the impact is 5.248 J.