A 0.1 kg arrow with an initial velocity of 30 m/s hits a 4.0 kg melon initially at rest on a friction-less surface. The arrow emerges out the other side of the melon with a speed of 20 m/s. What is the speed of the melon? Why would we normally not expect to see the melon move with the is speed after being hit by the arrow?
To find the speed of the melon after being hit by the arrow, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity. In this case, the momentum of the arrow before the collision is given by the product of its mass (0.1 kg) and initial velocity (30 m/s), which is equal to 0.1 kg × 30 m/s = 3 kg·m/s. Since the melon is initially at rest, its momentum before the collision is 0.
After the collision, the momentum of the arrow is given by the product of its mass (0.1 kg) and final velocity (20 m/s), which is equal to 0.1 kg × 20 m/s = 2 kg·m/s. Now, let's denote the speed of the melon as v (in m/s).
According to the conservation of momentum, the momentum after the collision should be equal to the initial momentum:
Momentum of arrow after collision + Momentum of melon after collision = Initial momentum
2 kg·m/s + (4 kg × v) = 3 kg·m/s
Rearranging the equation, we get:
4 kg × v = 3 kg·m/s - 2 kg·m/s
4 kg × v = 1 kg·m/s
Dividing both sides of the equation by 4 kg, we find:
v = 1 kg·m/s ÷ 4 kg
v = 0.25 m/s
Therefore, the speed of the melon after being hit by the arrow is 0.25 m/s.
In reality, we would not expect to see the melon move with this speed after being hit by the arrow because the melon is significantly more massive than the arrow. The melon has a mass of 4.0 kg, while the arrow only has a mass of 0.1 kg. Due to its larger mass, the melon would have a significantly smaller speed compared to the arrow after the collision. The arrow, on the other hand, experiences a larger change in momentum as it penetrates the melon, which causes it to come out with a greater speed relative to the melon.