A 20.0kg curling stone at the hog line and moves 28.35m [west] to sit on the button and score a point. If the coefficient of kinetic friction between the stone and the ice is 0.00200, what was the initial speed of the stone when it was released?
Well, let's put our thinking curlers on for this one! To find the initial speed of the stone, we can use the work-energy principle. The work done by the friction force is equal to the change in kinetic energy. And since the stone started from rest, the initial kinetic energy is zero.
First, let's find the work done by friction. The formula is W = F * d * cos(theta), where W is the work done, F is the force, d is the distance, and theta is the angle between the force and displacement. In this case, the force is the friction force, which is equal to the coefficient of kinetic friction multiplied by the normal force.
The normal force is equal to the weight of the stone, which is mass times gravity. So, the friction force is 0.00200 * (20.0kg * 9.8m/s²).
Now, we know the work done by friction is equal to the change in kinetic energy. Therefore, W = (1/2) * m * v², where W is the work done, m is the mass, and v is the final velocity. Since the stone ends up at rest, the final velocity is zero.
By equating the two equations, we have:
0.00200 * (20.0kg * 9.8m/s²) * 28.35m = (1/2) * 20.0kg * v²
Now, let's solve for v.
To find the initial speed of the stone when it was released, we can use the concept of work and energy.
The work done against friction is given by the formula:
Work = Force x Distance
The force of friction can be calculated using the formula:
Force of friction = coefficient of friction x Normal force
And the normal force is equal to the weight of the stone, which can be calculated using the formula:
Normal force = mass x gravity
Given:
Mass of the stone (m) = 20.0 kg
Coefficient of kinetic friction (μ) = 0.00200
Step 1: Calculate the normal force
Normal force = mass x gravity
Normal force = 20.0 kg x 9.8 m/s^2
Normal force = 196.0 N
Step 2: Calculate the force of friction
Force of friction = coefficient of friction x Normal force
Force of friction = 0.00200 x 196.0 N
Force of friction = 0.392 N
Step 3: Calculate the work done against friction
Work = Force x Distance
Work = 0.392 N x 28.35 m
Work = 11.1072 N·m or Joules
Step 4: Calculate the initial kinetic energy of the stone
The work done against friction is equal to the loss in kinetic energy.
Initial kinetic energy - Work = Final kinetic energy
Since the stone starts at rest, the initial kinetic energy is zero.
0 - 11.1072 N·m = 0.5 x mass x (initial velocity)^2
Simplifying the equation:
-11.1072 N·m = 0.5 x 20.0 kg x (initial velocity)^2
Step 5: Calculate the initial velocity
(initial velocity)^2 = (-11.1072 N·m) / (0.5 x 20.0 kg)
(initial velocity)^2 = -0.55536 N·m / kg
(initial velocity)^2 = -27.768 m^2/s^2
Since velocity cannot be negative,
initial velocity = sqrt(-27.768 m^2/s^2) = 5.271 m/s
Therefore, the initial speed of the stone when it was released was 5.271 m/s.
To find the initial speed of the stone when it was released, we can use the principle of conservation of mechanical energy.
First, let's calculate the work done by friction on the curling stone as it moves. The work done by friction is given by the equation:
Work = Force_friction * Distance * cos(theta)
Where:
- Force_friction is the force of friction between the stone and the ice.
- Distance is the distance the stone moves along the ice.
- cos(theta) is the cosine of the angle between the force of friction and the direction of motion. In this case, the force of friction is in the opposite direction to the motion, so cos(theta) = -1.
Now, the work done by friction can be expressed as the change in the stone's kinetic energy:
Work = Change in Kinetic Energy = KE_final - KE_initial
Since the stone starts from rest, the initial kinetic energy (KE_initial) is zero. Therefore, the work done by friction is equal to the final kinetic energy (KE_final).
Next, let's calculate the work done by friction. The force of friction between the stone and the ice is given by the equation:
Force_friction = coefficient of kinetic friction * Normal force
The normal force is the force exerted by the ice on the stone in the upward direction, which is equal to the weight of the stone in this case.
Normal force = Mass * Acceleration due to gravity
Finally, we can calculate the initial speed of the stone using the formula for kinetic energy:
KE_final = (1/2) * Mass * Velocity^2
By substituting all these values and solving the equations, we can find the initial speed of the stone when it was released.
F = - 20 * 9.8 * .002
F = m a
-20 * 9.8 * .002 = 20 a
a = -19.6 *10^-3 m/s^2
stops at time t at the button
v = Vi + a t
0 = Vi + a t
t = -Vi/a
d = Vi t + (1/2) a t^2
28.35 = Vi (-Vi/a) + (a/2) Vi^2/a^2
28.35 = -Vi^2/a + (1/2) Vi^2/a
28.35 = -(1/2) Vi^2/a
so
Vi^2 = -2 (28.35)(-19.6*10^-3)
Vi^2 = 1111
Vi = 33.3 m/s