A catapult is launching rocks at your village. It is hiding in a forest. All you can measure is that
the rock just passes over an 8 meter tall tree that is 73 meters from the village and that it takes 3
seconds for the rock to arrive from that moment.
How far is the catapult from the village?
No Idea if I am doing this right.
But so far what I done was
10.5Sin(67.6) = - gΔt
and got -9.71
u = horizontal speed = 73/3 = 24.3 m/s
falls 8 meters in 3 seconds
h = Hi + Vi t - 4.9 t^2
0 = 8 + 3 Vi - 9(4.9)
Vi = 12 m/s upward speed at tree top of 8 m
so let's find the top above that tree
v = Vi -9.81 t
0 = 12 - 9.81 t
t = 1.22 seconds rising above tree
so
h at top
h = 8 + 12(1.22) - 4.9 (1.22)^2
h = 15.3 meters high at top
so
falls to ground from 15.3 meters
How long to fall?
h = (1/2) g t^2
15.3 = 4.9 t^2
t = 1.77 seconds in the air falling
same time rising
so
total time in air = 3.54 seconds at horizontal speed of 24.3 m/s
so
answer is 3.54 * 24.3 = 86 meters
To determine the distance between the catapult and the village, we can use the principles of projectile motion.
First, let's identify the known information:
- The height of the tree: 8 meters
- The horizontal distance between the tree and the village: 73 meters
- The time it takes for the rock to arrive after passing over the tree: 3 seconds
To find the distance between the catapult and the village, we need to use the equation for the horizontal motion of a projectile. This equation is given by:
Distance = Initial velocity * Time
In this case, the initial velocity in the horizontal direction is constant since there is no acceleration acting in that direction. Therefore, we only need the time taken for the projectile to travel.
Using the given time (3 seconds) and horizontal distance (73 meters), we can calculate the initial velocity in the horizontal direction. Rearranging the equation, we have:
Initial velocity = Distance / Time
Substituting the values, we get:
Initial velocity = 73 meters / 3 seconds
Simplifying, we find that the initial velocity in the horizontal direction is:
Initial velocity = 24.333 meters/second
Now that we have the initial velocity, we can use this information to determine the distance between the catapult and the village. Since the vertical motion of the projectile is unaffected by the horizontal motion, we can use the equation for the vertical motion to find the distance traveled in the vertical direction using the known height (8 meters) and the initial vertical velocity.
The equation for the vertical motion of a projectile is:
Distance = (Initial vertical velocity * Time) + (0.5 * Acceleration * Time^2)
However, since we are only interested in the distance traveled at the moment the rock passes over the tree, we can ignore the second term on the right side of the equation. This is because the vertical distance traveled due to acceleration is negligible over such a short time span.
Therefore, the equation becomes:
Distance = Initial vertical velocity * Time
Now, we need to find the initial vertical velocity of the rock. To do this, we can use the time it takes for the rock to arrive (3 seconds) and the known height of the tree (8 meters).
Using the equation:
Distance = Initial vertical velocity * Time
And substituting the values:
8 meters = Initial vertical velocity * 3 seconds
Simplifying, we find that the initial vertical velocity is:
Initial vertical velocity = 8 meters / 3 seconds
Initial vertical velocity = 2.667 meters/second
Now that we have both the initial vertical velocity (2.667 meters/second) and the initial horizontal velocity (24.333 meters/second), we can determine the distance between the catapult and the village using the Pythagorean theorem.
The Pythagorean theorem states:
Hypotenuse^2 = Base^2 + Height^2
In our case, the hypotenuse represents the distance between the catapult and the village, the base represents the horizontal distance (73 meters), and the height represents the vertical distance (8 meters).
Plugging in the values:
Distance^2 = (73 meters)^2 + (8 meters)^2
Simplifying:
Distance^2 = 5329 meters^2 + 64 meters^2
Distance^2 = 5393 meters^2
Taking the square root of both sides:
Distance = √(5393 meters^2)
Calculating this, we find that the distance between the catapult and the village is approximately:
Distance ≈ 73.44 meters
Therefore, the catapult is approximately 73.44 meters away from the village.