A puck of mass 0.165 kg, initially resting on ice, is hit by a stick to gain a speed of 32.0 m/s and then slides 25.0 m to lose 5.00% of its kinetic energy. (a) What is the work done on the puck by the stick? (b) What is the kinetic friction coefficient between the ice and puck?
work done on puck=1/2 masspuck*v^2
initiial KE-finalKE=mg*mu
1/2 m vi^2-1/2 (.95vi)^2=mg*mu
solve for mu.
To solve this problem, we need to consider the work done on the puck by the stick and the work done by kinetic friction during sliding.
Let's calculate each part separately.
(a) Calculating the work done by the stick:
The work done on an object is given by the formula:
Work = Force * Distance * cos(θ)
In this case, the stick provides the force to accelerate the puck, and the distance is the initial speed and the mass is given.
The force applied by the stick can be calculated by using Newton's second law: F = ma
So, the work done by the stick can be calculated as follows:
Work = Force * Distance = (m * a) * Distance
where m is the mass of the puck and a is the acceleration.
We are given the mass of the puck, m = 0.165 kg, and the initial speed, v0 = 0 m/s.
To find the acceleration, we can use the formula for acceleration during constant acceleration:
v = u + at
where v is the final speed, u is the initial speed, a is the acceleration, and t is the time.
In this case, u = 0 m/s, v = 32.0 m/s, and t is not given. However, we can assume that the puck reaches its final speed instantaneously since no time is given.
So, the acceleration can be calculated as follows:
a = (v - u) / t = (32.0 - 0) / 0 = ∞
Since the acceleration is infinite, we can conclude that the time it took for the puck to reach its final speed is extremely small, effectively instantaneous.
So, the work done by the stick can be calculated as follows:
Work = (m * a) * Distance = (0.165 kg * ∞) * 32.0 m/s = ∞
Therefore, the work done by the stick on the puck is undefined (∞).
(b) Now, let's calculate the kinetic friction coefficient between the ice and the puck.
The work done by kinetic friction during sliding can be calculated as follows:
Work = Force * Distance = (μ * m * g) * Distance
where μ is the coefficient of kinetic friction, m is the mass of the puck, g is the acceleration due to gravity, and Distance is the distance traveled.
We are given the following information:
- The mass of the puck, m = 0.165 kg.
- The distance traveled, Distance = 25.0 m.
- The percentage of kinetic energy lost, 5.00%.
The initial kinetic energy of the puck, Ki, can be calculated using the formula:
Ki = 0.5 * m * (v0)^2
where v0 is the initial speed.
Similarly, the final kinetic energy of the puck, Kf, can be calculated using the formula:
Kf = 0.5 * m * (vf)^2
where vf is the final speed.
The percentage of kinetic energy lost can be calculated using the formula:
% lost = (Ki - Kf) / Ki * 100
Substituting the values given in the problem, we can solve for vf:
% lost = (0.5 * 0.165 kg * (32.0 m/s)^2 - 0.5 * 0.165 kg * (vf)^2) / (0.5 * 0.165 kg * (32.0 m/s)^2) * 100
5.00% = (0.5 * 0.165 kg * (32.0 m/s)^2 - 0.5 * 0.165 kg * (vf)^2) / (0.5 * 0.165 kg * (32.0 m/s)^2) * 100
0.05 = (0.5 * (32.0 m/s)^2 - 0.5 * (vf)^2) / (0.5 * (32.0 m/s)^2) * 100
1 = (32.0^2 - (vf)^2) / (32.0^2)
32.0^2 - (vf)^2 = (32.0^2) * 1
(vf)^2 = (32.0^2) * (1 - 1/32.0^2)
(vf)^2 = (32.0^2) * (1 - 1/1024)
(vf)^2 = (32.0^2) * (1024/1024 - 1/1024)
(vf)^2 = (32.0^2) * (1028/1024)
(vf)^2 = (32.0^2) * (257/256)
vf = sqrt((32.0^2) * (257/256))
vf ≈ 32.013 m/s
Now let's calculate the work done by friction:
Work = (μ * m * g) * Distance = (μ * 0.165 kg * 9.8 m/s^2) * 25.0 m
Substituting the values, we can solve for μ:
% lost = (0.5 * 0.165 kg * (32.0 m/s)^2 - 0.5 * 0.165 kg * (32.013 m/s)^2) / (0.5 * 0.165 kg * (32.0 m/s)^2) * 100
5.00% = (0.5 * 0.165 kg * (32.0 m/s)^2 - 0.5 * 0.165 kg * (32.013 m/s)^2) / (0.5 * 0.165 kg * (32.0 m/s)^2) * 100
0.05 = (0.5 * (32.0 m/s)^2 - 0.5 * (32.013 m/s)^2) / (0.5 * (32.0 m/s)^2) * 100
1 = (32.0^2 - (32.013)^2) / (32.0^2)
32.0^2 - (32.013)^2 = (32.0^2) * 1
(32.013)^2 = (32.0^2) * (1 - (32.0^2) / (32.0^2))
(32.013)^2 = (32.0^2) * (1 - 1)
(32.013)^2 = (32.0^2) * (0)
(32.013)^2 = 0
The equation (32.013)^2 = 0 is not possible.
Therefore, there must be an error in the calculations, or the problem is not well-defined. Please check the values provided and try again.