William Tell shoots an apple from his son's head.The speed of the 125-g arrow just before it strikes the apple is 25.0m/s, and the time of impact it is travelling horizontally.If the arrow strikes in the apple and the Arrow/Apple combination strikes 8.50m behind the son's feet, how massive was the apple? Assume the son is 1.85m tall
.125*25=total mass*combined velocity
=M+.125)V
The time to fall off the head is
1.85=9.8t^2/2
solve for time,t
then
V*t=8.5 solve for V
Put that back into the first equation, solve for M
Well, it seems that William Tell was aiming for quite the "apple-ling" experience! Let's see if we can help out here.
To solve this problem, we can start by considering the vertical motion of the arrow. We know that the arrow's initial velocity in the vertical direction is 0 m/s since it is only traveling horizontally. The time it takes for the arrow to reach the apple can be calculated using the following equation:
Δy = Vyi * t + (1/2) * (gravity) * t^2
Since Δy (the vertical displacement) is equal to the height of the son (1.85m), Vyi (initial vertical velocity) is 0 m/s, and we can approximate the acceleration due to gravity to be around 9.8 m/s^2. Solving for t gives us:
1.85 = 0 * t + (1/2) * 9.8 * t^2
Simplifying the equation, we get:
4.9 * t^2 = 1.85
Now let's calculate the time it takes for the arrow to reach the apple. Using this time value, we can then find the horizontal displacement of the arrow.
Okay, enough with the math! Now, let's talk about the apple's mass. Since we know the horizontal displacement of the arrow/apple combination is 8.50m, we can work backwards to determine the apple's mass.
Assuming that the collision between the arrow and the apple is perfectly elastic (which is highly unlikely, but let's go with it), we can utilize the conservation of momentum. The equation goes like this:
(marrow * varrow_initial) = (mapple * vapple)
"m" represents mass, and "v" represents velocity. We already know the mass of the arrow (125g) and its initial velocity (25.0 m/s).
Rearranging the equation, we can solve for the mass of the apple:
mapple = (marrow * varrow_initial) / vapple
Now, we just need to calculate the velocity of the apple. Since the arrow is moving horizontally, it imparts its entire horizontal velocity to the apple at the time of collision. Therefore, vapple will simply be the horizontal velocity of the arrow just before impact (25.0 m/s).
Plug in the values, and you'll find the mass of the apple. Voila!
To find the mass of the apple, we can use the principle of conservation of momentum. The momentum before the arrow strikes the apple is equal to the momentum after the arrow strikes the apple.
The momentum before can be calculated as the product of the mass of the arrow and its velocity:
Momentum before = mass of arrow × velocity of arrow
Given:
Mass of the arrow (m1) = 125 g = 0.125 kg
Velocity of the arrow (v1) = 25.0 m/s
Momentum before = 0.125 kg × 25.0 m/s
Next, let's consider the momentum after the arrow strikes the apple. The apple and the arrow will be traveling together with the same velocity. To find the velocity, we can use the equation for horizontal distance:
Distance = velocity × time
Given:
Distance = 8.50 m
Velocity = velocity after arrow strikes apple (v2)
Time = time of impact
To find the time of impact, we can use the equation for vertical distance:
Distance = (1/2) × gravity × time^2
Given:
Distance = height of the son = 1.85 m
Gravity = 9.8 m/s^2 (acceleration due to gravity)
Time = time of impact (t)
Solving for time:
1.85 m = (1/2) × 9.8 m/s^2 × t^2
Simplifying:
t^2 = (2 × 1.85 m) / (9.8 m/s^2)
t^2 = 0.3776 s^2
t ≈ 0.615 s
Now, let's find the velocity after the arrow strikes the apple:
Distance = velocity × time
8.50 m = v2 × 0.615 s
v2 = 8.50 m / 0.615 s
v2 ≈ 13.821 m/s
Now, we can find the momentum after the arrow strikes the apple using the mass of the arrow and the velocity of the arrow and the apple:
Momentum after = (mass of arrow + mass of apple) × velocity after
Since the arrow and the apple are traveling together after the impact, we can rewrite this equation as:
Momentum after = (mass of arrow + mass of apple) × velocity after = Momentum before
(0.125 kg + mass of apple) × 13.821 m/s = 0.125 kg × 25.0 m/s
0.125 kg + mass of apple = (0.125 kg × 25.0 m/s) / 13.821 m/s
mass of apple ≈ (0.125 kg × 25.0 m/s) / 13.821 m/s - 0.125 kg
mass of apple ≈ 0.227 kg
Therefore, the mass of the apple is approximately 0.227 kg.
To find the mass of the apple, we can use the principles of conservation of momentum.
The momentum of an object is given by the product of its mass and velocity: p = m * v.
First, let's find the momentum of the arrow before impact. Given its mass (m) of 125 g (which is equal to 0.125 kg) and velocity (v) of 25.0 m/s, we can calculate its momentum:
p_arrow = m_arrow * v_arrow
= 0.125 kg * 25.0 m/s
= 3.125 kg·m/s
Since momentum is conserved, the total momentum before impact should be equal to the total momentum after impact. Initially, only the arrow is moving, so the initial momentum is the momentum of the arrow alone: p_initial = p_arrow.
After impact, the arrow and the apple combination will be moving together. The momentum of the arrow/apple combination just before striking the ground (assuming it strikes directly behind the son's feet) will be equal to the initial momentum. We can write this as:
p_final = p_initial
= p_arrow
Now, let's assume the apple has a mass of m_apple. We know that the velocity of the arrow/apple combination just before impact is 25.0 m/s, since it is already given.
The momentum of the arrow/apple combination just before impact is the sum of the momentum of the arrow and the momentum of the apple:
p_final = p_arrow + p_apple
= m_arrow * v_final + m_apple * v_final
Since the arrow and the apple combination move together just before impact, their final velocity (v_final) is the same.
Substituting the known values, we have:
3.125 kg·m/s = 0.125 kg * 25.0 m/s + m_apple * 25.0 m/s
Simplifying:
3.125 kg·m/s = 3.125 kg·m/s + 25.0 m/s * m_apple
Subtracting 3.125 kg·m/s from both sides:
0 = 25.0 m/s * m_apple
Since we have multiplied both sides by a positive number (25.0 m/s), we can divide both sides by 25.0 m/s:
0 / 25.0 m/s = m_apple
m_apple = 0
This result seems odd because it implies that the mass of the apple is zero. However, we need to consider the assumption that the arrow and the apple combination strike the ground directly behind the son's feet. In reality, any deviation from this assumption would result in a different outcome.
So, based on the given information and the assumptions made, it appears that the mass of the apple cannot be determined using conservation of momentum alone.