A 4.5 g dart is fired into a block of wood with a mass of 23.8 g. The wood block is initially at rest on a 1.3 m tall post. After the collision, the wood block and dart land 3.6 m from the base of the post. Find the initial speed of the dart.
To find the initial speed of the dart, we can use the conservation of momentum.
The momentum before the collision (dart + wood block) is equal to the momentum after the collision.
The formula for momentum is:
momentum = mass x velocity
Let's assign variables to the given values:
mass of the dart (m1) = 4.5 g = 0.0045 kg
mass of the wood block (m2) = 23.8 g = 0.0238 kg
height of the post (h) = 1.3 m
distance from the base of the post (d) = 3.6 m
Before the collision, both the dart and the wood block are at rest. Therefore, the initial momentum (momentum before the collision) is zero.
After the collision, both the dart and the wood block land 3.6 m from the base of the post, which means their horizontal velocity is the same.
Let v be the initial velocity of the dart.
The momentum after the collision is:
momentum = (mass of the dart x velocity) + (mass of the wood block x velocity)
momentum = (m1 + m2) x velocity
Since momentum is conserved, we have:
0 = (m1 + m2) x velocity
Substituting the given values:
0 = (0.0045 kg + 0.0238 kg) x velocity
Simplifying the equation:
0 = 0.0283 kg x velocity
To solve for velocity, we divide both sides of the equation by 0.0283 kg:
velocity = 0 / 0.0283 kg
The velocity is zero, which means the initial speed of the dart is also zero.
Therefore, the initial speed of the dart is 0 m/s.
To find the initial speed of the dart, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision, assuming no external forces act on the system.
The momentum of an object is defined as the product of its mass and velocity. So, we can express the initial momentum of the dart-wood system as:
Initial momentum = (mass of dart) × (velocity of dart) + (mass of wood block) × (velocity of wood block before collision)
Since the wood block is initially at rest, its velocity before the collision is 0. Therefore, the initial momentum simplifies to:
Initial momentum = (mass of dart) × (velocity of dart)
Similarly, the final momentum of the system can be expressed as:
Final momentum = (mass of dart) × (velocity of dart) + (mass of wood block) × (velocity of wood block after collision)
After the collision, the dart and wood block land 3.6 m from the base of the post, so they have the same final velocity. Therefore, we can rewrite the final momentum as:
Final momentum = (mass of dart) × (velocity of dart) + (mass of wood block) × (velocity of dart)
Since the total momentum before and after the collision is the same, we can equate the initial and final momentum expressions:
(mass of dart) × (velocity of dart) = (mass of dart) × (velocity of dart) + (mass of wood block) × (velocity of dart)
Simplifying the equation:
0 = (mass of wood block) × (velocity of dart)
From the equation, we can see that the mass of the wood block is not relevant to finding the initial speed of the dart. The equation states that the velocity of the dart is 0, which means the dart is at rest initially. However, this contradicts the question statement that the dart is fired into the wood block.
Therefore, there might be an error in the given information or calculation. Please double-check the question and make sure all the values are accurate to proceed with finding the initial speed of the dart.