An ice skater at rest on ice catches a dance partner moving 2.5 m/s during a performance. The ice skater has a mass of 68 kg and the dance partner has a mass of 54 kg. What is the speed of the ice skater and dancer partner after the collision?
A) 2.3 m/s
B) 2.0 m/s
C) 1.4 m/s
D) 1.1 m/s
A machine works with an efficiency of 18%. The energy input is 500 joules/second. How much useful energy will the machine generate in 15 minutes?
A) 81 kg
B) 1.35 kg
C) 25 kg
D) 135 kg
A lever is being used to try to lift a heavy object but it is not working. Which of these actions would increase the output force the most?
A) decrease the input force and increase the output displacement
B) increase the input force and decrease the output displacement
C) decrease the input force and decrease the output displacement
D) increase the input force and increase the output displacement
Before I give you the correct answers, let me entertain you with a little joke.
Why couldn't the bicycle stand up by itself?
Because it was two-tired!
Now, on to the serious business of answering your questions.
For the first question, we can use the principle of conservation of momentum to find the final speed. Since the ice skater is at rest, the total initial momentum is zero. After the collision, the total final momentum must also be zero. This means the ice skater and the dance partner will move in opposite directions with equal magnitudes of momentum. By using the law of conservation of momentum, we can calculate the final speed of the ice skater and dancer partner to be 2.5 m/s. Therefore, the correct answer is not on the list because it should be 2.5 m/s.
Now, let's move on to the second question. The efficiency of a machine is calculated by dividing the useful energy output by the total energy input. In this case, with an efficiency of 18%, the machine will generate 18% of the total energy input as useful energy. The total energy input is given as 500 joules/second. Therefore, the machine will generate (18/100) * 500 = 90 joules of useful energy every second. In 15 minutes (or 900 seconds), the machine will generate 900 * 90 = 81,000 joules of useful energy. So, the correct answer is A) 81 kg.
Finally, let's tackle the third question. To increase the output force of a lever, we need to either increase the input force or increase the lever arm (output displacement). So, the correct answer is D) increase the input force and increase the output displacement.
I hope my humorous explanations brought a smile to your face. If you have any more questions, I'm here to clown around and help!
To solve the first question, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
Let's define the initial momentum of the ice skater as m1v1 and the initial momentum of the dance partner as m2v2.
Before the collision, the ice skater is at rest, so the initial momentum of the ice skater is 0 (m1v1 = 0).
The total initial momentum before the collision is therefore m2v2.
After the collision, we can assume the ice skater and the dance partner move together with a common velocity, let's call it vf.
The total momentum after the collision is (m1 + m2) * vf.
According to the conservation of momentum, m2v2 = (m1 + m2) * vf.
Now, we can solve for vf:
vf = m2v2 / (m1 + m2)
Plugging in the given values:
m1 = 68 kg
m2 = 54 kg
v2 = 2.5 m/s
vf = (54 kg * 2.5 m/s) / (68 kg + 54 kg)
vf = 135 kg·m/s / 122 kg
vf ≈ 1.1 m/s
Therefore, the speed of the ice skater and dance partner after the collision is approximately 1.1 m/s.
Answer for the first question is D) 1.1 m/s.
Now let's move to the second question.
To find the useful energy generated by the machine, we need to calculate the energy output of the machine.
The efficiency of the machine is given as 18%. This means that the machine only converts 18% of the energy input into useful energy. The remaining energy is wasted as heat or other forms of energy.
The energy output of the machine can be calculated by multiplying the energy input by the efficiency:
Energy output = Energy input * Efficiency
Given:
Energy input = 500 joules/second
Efficiency = 18% = 0.18
Energy output = 500 joules/second * 0.18 = 90 joules/second
Now we need to calculate the total useful energy generated by the machine in 15 minutes.
Since there are 60 seconds in a minute, and 15 minutes is equivalent to 15 * 60 = 900 seconds.
Total useful energy generated = Energy output * Time
Total useful energy generated = 90 joules/second * 900 seconds = 81,000 joules
Therefore, the machine will generate 81,000 joules of useful energy in 15 minutes.
Answer for the second question is A) 81 kg.
Now let's move to the third question.
To increase the output force the most, we need to increase the mechanical advantage of the lever system.
From the given options, the action that would achieve this is:
D) Increase the input force and increase the output displacement.
By increasing the input force and the output displacement, the mechanical advantage of the lever system will increase, resulting in an increased output force.
Answer for the third question is D) Increase the input force and increase the output displacement.
To find the speed of the ice skater and dance partner after the collision, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
Here's how you can solve this problem step by step:
1. Identify the initial and final states:
- Initial state: Ice skater at rest on ice and dance partner moving at 2.5 m/s.
- Final state: Both ice skater and dance partner move with some final speed.
2. Write down the conservation of momentum equation:
- Initial momentum = Final momentum
The initial momentum is given by:
momentum of ice skater (initial) = 0 kg x m/s (since the ice skater is at rest)
momentum of dance partner (initial) = mass of dance partner x velocity of dance partner
The final momentum is given by:
momentum of ice skater (final) = mass of ice skater x velocity of ice skater (final)
momentum of dance partner (final) = mass of dance partner x velocity of dance partner (final)
3. Set up the equation and solve for the final velocity of the ice skater and dance partner:
- (0) + (54 kg x 2.5 m/s) = (68 kg x Vfinal) + (54 kg x Vfinal)
Simplifying and solving for Vfinal:
135 kg x m/s = 122 kg x Vfinal
Vfinal = 135 kg x m/s / 122 kg
Vfinal ≈ 1.11 m/s
So, the speed of the ice skater and dance partner after the collision is approximately 1.11 m/s.
Therefore, the correct answer is:
D) 1.1 m/s
To find the useful energy generated by a machine, we need to calculate the product of the energy input and the efficiency of the machine. Here's how you can solve the second problem step by step:
1. Calculate the useful energy generated by the machine:
- Useful energy = energy input x efficiency
2. Convert the time from minutes to seconds:
- 15 minutes = 15 minutes x 60 seconds / 1 minute = 900 seconds
3. Calculate the useful energy in Joules:
- Useful energy = (energy input per second) x (time in seconds)
- Useful energy = 500 J/s x 900 s = 450,000 J
So, the machine will generate 450,000 Joules of useful energy in 15 minutes.
Therefore, the correct answer is:
D) 135 kg
To increase the output force the most when using a lever, we need to consider the concepts of force and displacement in relation to the lever arm. Here's how you can solve the third problem step by step:
1. Understand the lever principle:
- Levers work based on the principle of "force multiplication." By applying a relatively small input force over a large displacement, we can generate a much larger output force over a shorter displacement.
2. Based on the lever principle, we can determine the correct action to increase the output force the most:
- To maximize the output force, we need to increase the input force and increase the output displacement.
So, the correct answer is:
D) increase the input force and increase the output displacement