A 4.3 g dart is fired into a block of wood with a mass of 22.6 g. The wood block is initially at rest on a 1.4 m tall post. After the collision, the wood block and dart land 3.1 m from the base of the post. Find the initial speed of the dart.
m1 =4.3•10^-3 kg, m2 = 22.6•10^-3 kg, h =1.4 m, L= 3.1 m., v1 =?
m1 •v1 = (m1+m2) •v,
v= m1 •v1 /(m1 + m2) .
h =g•t²/2 => t = sqrt(2•h/g)
L= v•t = v• sqrt(2•h/g),
v =L/sqrt(2•h/g),
m1 •v1 /(m1 + m2) = L/sqrt(2•h/g).
v1= L•(m1 + m2)/m1• sqrt(2•h/g) = …
To find the initial speed of the dart, we can apply the principle of conservation of momentum. This principle states that the total momentum before an event is equal to the total momentum after the event, assuming there are no external forces acting on the system.
In this case, the dart and the wood block make up the system. Initially, the wood block is at rest, so its initial momentum is zero. The dart has an unknown initial speed v1 and a mass of 4.3 g.
After the collision, both the dart and the wood block land on the ground together. To find the final momentum, we need to find the final velocity of the system after the collision. We can do this by calculating the height from which the system falls, using the equation h = gt^2/2, where h is the height, g is the acceleration due to gravity (approximately 9.8 m/s^2), and t is the time of flight.
In this case, the height of the post is 1.4 m. The time of flight can be calculated using the equation t = sqrt(2h/g).
Plugging in the values, we find:
t = sqrt(2 * 1.4 m / 9.8 m/s^2)
t ≈ 0.53 seconds
Now, using the horizontal distance traveled (3.1 m) and the time of flight, we can calculate the final velocity using the equation v = d/t, where v is the final velocity, d is the distance, and t is the time.
v = 3.1 m / 0.53 s
v ≈ 5.85 m/s
Now, we can use the conservation of momentum equation to relate the initial and final momentum:
m1 * v1 = m2 * v2
where m1 is the mass of the dart (4.3 g), v1 is the initial velocity of the dart, m2 is the mass of the wood block (22.6 g), and v2 is the final velocity of the system (5.85 m/s).
To make the units consistent, we need to convert the masses from grams to kilograms:
m1 = 4.3 g = 0.0043 kg
m2 = 22.6 g = 0.0226 kg
Now we can substitute the values into the equation:
0.0043 kg * v1 = 0.0226 kg * 5.85 m/s
Solving for v1, we find:
v1 = (0.0226 kg * 5.85 m/s) / 0.0043 kg
v1 ≈ 30.82 m/s
Therefore, the initial speed of the dart is approximately 30.82 m/s.