Disappointed with a 4th place tin medal in bobsledding, Burl has decided to take up curling. One of the opposing team's stones is resting on the outer edge of the house as shown below. With which speed should Burl hit the other team's rock in order to knock it out of the way and land in the middle of the target?
The target has a radius(R) of 3.65m (the rock happens to be lying such that x and y are equal distances) and the stones each have a mass(m) of 17kg. The coefficient of friction between the ice and granite stones is 0.012. Assume that Burl is able to hit the stone at the correct angle to put his rock on a straight line course to the bullseye and that his incoming stone will slide completely in the y-direction as shown.Also, assume an elastic collision.
y
|
|______x
Target=O
X
O-------
\ | Y
R \ |
o opponent's stone
o burl's stone
Problem looks like this ibb.co/6nHQMfm
To determine the speed at which Burl should hit the opponent's stone, we can use the principle of conservation of momentum and the laws of motion. Here's how you can solve the problem step by step:
1. Understand the problem: In this scenario, Burl wants to hit the opponent's stone and land his own stone in the middle of the target (bullseye). We need to calculate the speed at which Burl should hit the opponent's stone to achieve this.
2. Identify the relevant equations: We'll be using the equations related to conservation of momentum and the laws of motion, specifically the equation for linear momentum (p = m * v) and Newton's second law (F = μ * N).
3. Analyze the forces acting on the opponent's stone: Since the opponent's stone is lying on the ice, the only external force acting on it is the force of friction. Friction opposes the motion and is given by the equation F = μ * N, where μ is the coefficient of friction and N is the normal force.
4. Calculate the normal force: The normal force (N) is equal to the weight of the stone (mg) acting perpendicular to the surface of the ice. Since the stone is lying on its edge, the normal force is equal to the weight of half the stone, which is (1/2) * mg.
5. Calculate the force of friction: With the normal force determined, we can calculate the force of friction using the equation F = μ * N.
6. Consider the motion of the opponent's stone: Since Burl wants his own stone to hit the opponent's stone and move only in the y-direction (straight towards the bullseye), the opponent's stone must move only in the x-direction. Therefore, the force of friction acting on the opponent's stone will only oppose its x-direction motion.
7. Apply Newton's second law in the x-direction: Since the only force acting on the opponent's stone is friction, we can use Newton's second law (F = ma) to find the acceleration in the x-direction. The mass cancels out, giving us a = F / m.
8. Solve for acceleration in the x-direction: Plug in the force of friction (F) and the mass of the opponent's stone (m) into the equation to calculate the acceleration in the x-direction.
9. Use kinematic equations to calculate the time it takes for the opponent's stone to reach the bullseye: Since we know the initial speed of the opponent's stone is 0, and its final position is at the center of the target (x = 0), we can use the kinematic equation x = 0.5 * at^2 to solve for the time (t) it takes for the stone to move from its initial position to the target.
10. Calculate the distance Burl's stone needs to travel: Since Burl wants his stone to land in the middle of the target, the distance it needs to travel is equal to the radius of the target (3.65m).
11. Use the equation of motion to calculate the required speed: Using the equation of motion d = vt, where d is the distance and t is the time, you can calculate the speed (v) that Burl's stone needs to have to reach the target distance within the calculated time.
12. Calculate the impulse required: Finally, we can calculate the impulse required to change the velocity of the opponent's stone in the x-direction to zero, using the equation impulse = m * Δv. Since we know the mass of the opponent's stone (m) and the initial and final velocity in the x-direction (0 and Δv, respectively), we can calculate the impulse.
13. Apply the principle of conservation of momentum: With the impulse required known, we can now apply the principle of conservation of momentum. Since the collision is elastic, the total momentum before the collision is equal to the total momentum after the collision. Since both stones have the same mass, the magnitude of the opposing stone's final velocity should be equal to Burl's initial velocity to ensure momentum conservation.
By following these steps and performing the necessary calculations, you should be able to determine the required speed at which Burl should hit the opponent's stone to knock it out of the way and land his stone in the middle of the target.