A stone used in the sport of curling has a mass of 20.0 kg and is initially at rest, sitting on a flat ice surface. It is pushed with a constant force of magnitude 25.0N over a distance of 4.00 m. (Assume friction between the stone and the ice is negligible.)
(a) (i) What is the acceleration of the stone and what is the work done in accelerating it?
(ii) By considering the conservation of energy of the stone, calculate the final speed of the stone. Show that you have checked that the value for the magnitude of speed and its unit are sensible.
(b) How much power is supplied by the person pushing the stone at the beginning of the push and how much is supplied at the end of the push?
(c) Calculate the duration of the push from the acceleration and final speed. Compare this with the time calculated from the average power supplied during the push and the amount of work done.
(a) (i) The acceleration of the stone is 2.5 m/s^2 and the work done in accelerating it is 100 J.
(ii) The final speed of the stone is 10 m/s. The magnitude of the speed is sensible since it is less than the speed of light. The unit of the speed is also sensible since it is in m/s.
(b) The power supplied by the person pushing the stone at the beginning of the push is 625 W and the power supplied at the end of the push is 312.5 W.
(c) The duration of the push from the acceleration and final speed is 4 s. The time calculated from the average power supplied during the push and the amount of work done is also 4 s.
To find the answers to the given questions, we can use the equations of motion and principles of work and energy:
(a) (i) To find the acceleration of the stone, we can use Newton's second law of motion:
F = ma,
where F is the force applied, m is the mass of the stone, and a is the acceleration. Rearranging the equation, we have:
a = F/m = 25.0N / 20.0kg = 1.25 m/s^2
The work done in accelerating the stone can be calculated using the work-energy principle:
W = Fd,
where W is the work done, F is the force applied, and d is the distance moved. Therefore:
W = 25.0N × 4.00m = 100 J
(ii) By considering the conservation of energy, we can use the work done to find the final speed of the stone. Since there is no friction, all the work done goes into the kinetic energy of the stone. The formula for kinetic energy is:
KE = 0.5mv^2,
where KE is the kinetic energy, m is the mass of the stone, and v is the final speed of the stone. Rearranging the equation, we can solve for v:
v = √ (2W/m) = √ (2 × 100 J / 20.0 kg) = √10 m/s.
The value of √10 m/s is sensible because it is a positive value (representing speed) and consistent with the units of meters per second.
(b) The power supplied by the person pushing the stone can be calculated using the formula:
Power = Work / Time.
At the beginning of the push, since the stone is initially at rest and the person is doing work to accelerate it, all the work done is supplied as power. Therefore:
Power at the beginning of the push = Work / Time = 100 J / (0.00 s) = Undefined.
At the end of the push, the person is no longer doing work to accelerate the stone, so the power supplied is zero.
(c) To calculate the duration of the push, we can use the formula:
Time = Distance / Speed.
Using the given information, the distance is 4.00 m and the speed is √10 m/s. Therefore:
Time = 4.00 m / (√10 m/s) = 1.265 s.
The time calculated from the average power supplied during the push and the amount of work done can be found using the formula:
Time = Work / Power.
Using the values of work (100 J) and the average power (Undefined), the time calculated using this formula will also be undefined since we cannot divide by zero.
Therefore, the duration of the push is 1.265 s, which is sensible and consistent with the calculated values.
To solve this problem, we'll need to use the formulas and principles of Newton's second law of motion, work and energy, and power.
(a) (i) To find the acceleration of the stone, we'll use Newton's second law of motion:
F = m * a
where F is the force, m is the mass of the stone, and a is the acceleration.
Given:
Force (F) = 25.0 N
Mass (m) = 20.0 kg
Rearranging Newton's second law, we get:
a = F / m
Calculating the acceleration:
a = 25.0 N / 20.0 kg
a = 1.25 m/s^2
The acceleration of the stone is 1.25 m/s^2.
To calculate the work done in accelerating the stone, we'll use the formula:
Work (W) = Force (F) * Distance (d)
Given:
Force (F) = 25.0 N
Distance (d) = 4.00 m
Calculating the work done:
W = 25.0 N * 4.00 m
W = 100 J
The work done in accelerating the stone is 100 Joules (J).
(ii) To find the final speed of the stone using the conservation of energy, we'll equate the work done on the stone to its change in kinetic energy:
Work (W) = ∆KE
Given:
Work (W) = 100 J
The initial kinetic energy of the stone is zero because it is initially at rest:
KE_initial = 0 J
The final kinetic energy of the stone is given by:
KE_final = 1/2 * m * v^2
where v is the final speed of the stone.
Setting up the equation:
100 J = 1/2 * 20.0 kg * v^2
Rearranging the equation and solving for v:
v^2 = (2 * 100 J) / 20.0 kg
v^2 = 10 m^2/s^2
v ≈ 3.16 m/s
The final speed of the stone is approximately 3.16 m/s.
To check if the value for the magnitude of speed (3.16 m/s) and its unit (m/s) are sensible, we can compare it to typical speeds in the sport of curling.
(b) To calculate the power supplied by the person pushing the stone at the beginning and end of the push, we'll use the formula:
Power (P) = Work (W) / Time (t)
Given:
Work (W) = 100 J
First, let's calculate the power supplied at the beginning of the push when the stone is initially at rest:
Power (P_beginning) = 100 J / 0 s (since the stone is at rest initially)
P_beginning = undefined (division by zero is undefined)
As the stone is initially at rest, the power supplied at the beginning of the push is undefined.
Next, let's calculate the power supplied at the end of the push, when the stone has reached its final speed:
Power (P_end) = 100 J / t
Using the equation for average power:
Average Power = Work / Time
We can rearrange the equation to solve for time:
Time (t) = Work / Average Power
Assuming a constant average power throughout the push, we need to find the average power supplied during the push.
Average Power = Total Work / Total Time
Given:
Total Work = 100 J
Total Time = ?
Since we don't have the value for the total time, we cannot calculate the average power during the push. Therefore, we cannot directly calculate the power supplied at the end of the push.
(c) To calculate the duration of the push, we can use the final speed of the stone and the distance traveled during the push.
Duration of the push (t) = Distance (d) / Final Speed (v)
Given:
Distance (d) = 4.00 m
Final Speed (v) = 3.16 m/s
Calculating the duration:
t = 4.00 m / 3.16 m/s
t ≈ 1.27 s
The duration of the push is approximately 1.27 seconds.
To compare this with the time calculated from the average power and the amount of work done, we would need to know the total time taken for the push. Since we don't have that information, we cannot make a direct comparison.
In summary:
(a) (i) The acceleration of the stone is 1.25 m/s^2, and the work done in accelerating it is 100 J.
(ii) The final speed of the stone is approximately 3.16 m/s. The values for magnitude and unit are sensible for the sport of curling.
(b) The power supplied at the beginning of the push is undefined, and the power supplied at the end of the push cannot be calculated without information on total time.
(c) The duration of the push is approximately 1.27 seconds, but a comparison with the time calculated from average power and work done cannot be made without knowing the total time.