Your friend has decided to make some money during the town tombola by inventing a game of skill that can be installed. In the dame as she has developed it so far, the customer shoots a rifle at a 5.00 cm diameter target falling straight down. Anyone who hits the target in the center wins a stuffed animal. Each shot would cost 50 cents. The rifle would be mounted on a pivot 1.00 meter above the ground so that it can point in any direction at any angle. When shooting, the customer stands 100 meters from where the target would hit the ground if the bullet misses. At the instant that the bullet leaves the rifle (with a muzzle velocity of 1317 km/h, according to the manual) the target is released from its holder 7.00 meters above the ground. Your friend asks you to try out the game which she has set up on a farm outside of town. Before you fire the gun you calculate where you should aim.
I started with converting 1317 km/h to 36.58m/s
so this should be the initial velocity and the acceleration is -9.81 m/s^2
and then i thought to use x=(v+vo)/2*t
but i don't know how to go on...
please help...
To calculate where you should aim to hit the target in the center, you need to find the horizontal distance the bullet will travel while the target falls. Here's how you can proceed:
1. Find the time it takes for the bullet to reach the ground:
Using the initial velocity (v0 = 36.58 m/s) and the acceleration due to gravity (a = -9.81 m/s^2), you can use the kinematic equation:
y = v0t + (1/2)at^2
Since the target falls from a height of 7.00 meters above the ground, you can set y = -7.00 meters (negative because it's below the starting point) and solve for t.
-7.00 = 36.58t + (1/2)(-9.81)t^2
This equation is a quadratic equation, so you can solve it using the quadratic formula or by factoring. Once you find the positive value of t, you'll have the time it takes for the bullet to reach the ground.
2. Calculate the horizontal distance traveled by the bullet in that time:
Since the bullet travels horizontally at a constant velocity, you can use the formula:
x = v0x * t
where v0x is the horizontal component of the initial velocity.
To find v0x, you can use the fact that the initial velocity (v0) can be broken down into vertical (v0y) and horizontal (v0x) components.
Since the rifle can point in any direction at any angle, the vertical and horizontal components of the initial velocity are related by:
v0x = v0 * cos(θ)
v0y = v0 * sin(θ)
where θ is the angle at which the rifle is pointed.
Given that the muzzle velocity is 36.58 m/s, you can calculate v0x by taking the cosine of the angle at which the rifle is pointed.
3. Find the horizontal distance:
Substitute the values you've found: t (from step 1) and v0x (from step 2) into the equation:
x = v0x * t
This will give you the distance you need to aim horizontally to hit the target in the center.
Remember to convert distances from meters to centimeters to ensure your aim is accurate.