For a YouTube video, a son agrees to let his dad shoot an apple off his head with a crossbow. The mass of the arrow is 125 g. The mass of the apple is 101 g. Right before the arrow hits the apple, the arrow is traveling at 25.0 m/s horizontally. If the son is 1.85 m tall, how far behind the son did the arrow (stuck in the apple) land?
momentum is conserved
.125 kg * 25.0 m/s = (.125 kg + .101 kg) * v
solve for the horizontal velocity (v) of the arrow/apple
the flight time of the arrow/apple is the time it takes to fall 1.85 m
1.85 = 1/2 g t^2 ... solve for the flight time (t)
the distance, behind the son, of the arrow/apple landing is ... v * t
To solve this problem, we can consider the conservation of momentum.
1. First, we need to find the initial momentum of the arrow before it hits the apple. Momentum is given by the equation: momentum = mass × velocity. The mass of the arrow is 125 g or 0.125 kg, and the velocity is 25.0 m/s. So, the initial momentum of the arrow is 0.125 kg × 25.0 m/s = 3.125 kg·m/s.
2. Since momentum is conserved, the final momentum of the system (the arrow and the apple stuck in it) will be equal to the initial momentum. The mass of the apple is 101 g or 0.101 kg. Therefore, the momentum of the apple (after being struck by the arrow) will be 0.101 kg × V, where V is its final velocity.
3. The final velocity of the system can be calculated using the principle of conservation of momentum. We know that momentum is a vector quantity, so we need to consider both its magnitude and direction. In this case, the arrow is initially moving horizontally, so the horizontal component of its momentum is equal to the momentum of the system after the apple is stuck in the arrow.
4. To find the horizontal component of the momentum, we need to consider that momentum is given by the equation: momentum = mass × velocity. The mass of the system (arrow + apple) is the sum of their masses, which is 125 g + 101 g = 226 g, or 0.226 kg. The final velocity of the system will be equal to the horizontal velocity component of the arrow, which is 25.0 m/s.
5. Now, we can calculate the momentum of the system after the apple is struck. Momentum = mass × velocity. The mass is 0.226 kg, and the velocity is 25.0 m/s. Thus, the momentum of the system after the apple is stuck is 0.226 kg × 25.0 m/s = 5.65 kg·m/s.
6. Since momentum is conserved, the initial momentum (3.125 kg·m/s) is equal to the final momentum (5.65 kg·m/s). Therefore, we can equate these two expressions:
Initial momentum = Final momentum
3.125 kg·m/s = 5.65 kg·m/s
7. Now, we can solve for the final velocity of the apple (V):
0.125 kg × 25.0 m/s = 0.226 kg × V
3.125 kg·m/s = 0.226 kg × V
V = 3.125 kg·m/s / 0.226 kg
V = 13.84 m/s
8. Now that we have the final velocity of the apple (V), we can use it to calculate the distance behind the son that the arrow (with the apple) landed. Since the son is not moving horizontally, the horizontal distance traveled by the arrow equals the horizontal distance traveled by the apple.
We can use the equation: distance = velocity × time
The time it takes for the arrow to travel the distance equal to the son's height (1.85 m) would be the same as the time it takes for the apple to fall the same distance. We can find this time using the equation for free fall: distance = (1/2) × g × t^2, where g = 9.8 m/s^2.
1.85 m = (1/2) × (9.8 m/s^2) × t^2
t^2 = (2 × 1.85 m) / (9.8 m/s^2)
t^2 = 0.3776 s^2
Taking the square root of both sides:
t = 0.614 s
Now we can find the distance traveled by the apple (and arrow) during this time:
distance = velocity × time
distance = 13.84 m/s × 0.614 s
distance ≈ 8.50 m
Thus, the arrow (with the apple) landed approximately 8.50 meters behind the son.