Article status: Draft
Time Estimate for Reading: 30 min
Learning Objectives: Velocity, Accelration and Power
Effort Required: Medium
Pedagogy Model: Evolution, Formula Analysis, Inter-disciplinary
Prior Physics Concepts: displacement, velocity, acceleration, mass
Prior Math Tools: Secondary school level Arithmetic, geometry and algebra
We wish to analyse and predict the motion of objects. So far we have identified that the amount of motion can be measured using the concepts of displacement, velocity and acceleration. We also understood the significance of acceleration; the bridge between space, time and mass.
Just to refresh our memory, measurement of velocity requires two points. Measure the distance between them and divide by the time interval.
Similarly, measurement of acceleration requires 3 points. measure the velocity between points 1 and 2, 2 and 3. also clock the time interval between points 1 and 3. now acceleration can be calculated by taking the difference in velocities and dividing by the total time taken. By getting the 3 point closer and closer, we would get more accurate measurement of acceleration. Mathematically speaking we say acceleration at a point (we shall see how geometry helped newton to settle this issue). but in practice we need 3 points.
Now that we know how to measure, We know the effect. Having known the effect, we are interested in causing the effect. Let us try to understand the cause of motion; motion in a straight line with constant velocity or constant acceleration. For that we will have to add the concept of power to our vocabulary.
What is power?
Let us begin with the definition, take a real life example and try to understand it step by step.
Definition:
Power = mass of the object x required acceleration x velocity of the object. This might look incomprehensible in the beginning but it is rather a very simple concept. It would be all the more intriguing if you have come across the text book definition "power is rate of doing work or rate of spending energy". Similar to displacement, work and energy are abstract concepts. They cannot be experienced in real life. We wont need them for the time being. Let us defer a discussion on work and energy later. So all we need is power to cause motion.
Since we state that power is real, let us take a real life example of using a two wheeler. Say a bike. the bike may be a by cycle (mechanical power), gasoline power bike (thermal power) or an electric bike (Electric power). Notice that power can be provided from different sources.
Electric Power = Voltage x current
Thermal power = mass x Specific heat capacity x average temperature difference/second
Mechanical power (linear motion) = mass x acceleration x average velocity
Mechanical power (rotational motion) = 2 x pi x revs per second x mass x acceleration x radius
Do not bother if you have not understood the formulas. we will get to it shortly.
Real life example 1: Bike
State 1.Initially the bike is a rest (initial velocity is zero).
State 2: Mount on the bike and start the engine. Let us assume it has an automatic gear and the road is perfectly horizontal.
State 3:Turn the accelerator (throttle) handle and the bike starts moving. You experience a change in velocity as long as you keep turning the accelerator handle (within the limit of traffic and vehicle capability).
Note down the distance traveled and time taken to reach the required velocity
State 4: Once we reach the required velocity (we wont be using the word speed. speed is required only for academic purposes), notice that we will not be twisting the accelerator handle any more. may be if you have noticed carefully we will release it quite a bit once we reached the required velocity.
State 5: Once the destination is reached (we are considering only straight line motion for now), Notice the distance traveled and time taken at constant velocity. Now, we release the accelerator handle from current position, we will observe that the velocity automatically decreases. Again measure note the distance traveled and time taken to reach zero velocity. The rate of decreasing velocity is termed deceleration. Just the opposite of acceleration.
If acceleration or deceleration are not constant, we will observe jerk; and jerk is not a comfortable experience.
Let us now go on to calculate the power required at various states.
Consider mass of the bike (including the person) = m kg
State 3: Power should be equal to mass x required acceleration of the object x average velocity
therefore power required = m x a x d/t. and is measured in watts.
m x a is quite intuitive. but, we may ask why d/t?. The answer is, any displacement will take a certain amount of time. So to define power in a sentence, Power is a quantity which is a product of mass to be accelerated through space in a given amount of time. the given amount of time is one second. Observe that during an accelerated motion, the velocity will keep changing every second. So, we need to keep changing the power every second (reason for gradually twisting the throttle). we may say that we should provide a continuously varying power. notice that, if you happen to accelerate for a period of 2 secs, the power required in 1 sec is less that the one in the next second (velocity would have increased by now)
There are other simpler methods to calculate continuously varying power using abstract concepts like work and kinetic energy. the concept of kinetic energy had to wait for calculus and calculus was invented (though there are many others who have contributed) by newton and leibnitz. So, we shall defer it for now.
State 4: Now we need a constant velocity motion. We would be tempted to apply the same reasoning above and would arrive at a quantity m x v. But this is not so. Why?
We have friction and air drag acting in the opposite direction. They offer deceleration (the opposite of acceleration) and so we cannot use v. How to calculate friction and air drag will be covered in a separate article. for now, it is sufficient to know that they offer deceleration.
Now the formula for power = mass x external deceleration x velocity
Notice that the required acceleration of the object has become equal to external deceleration. We just need to keep accelerating through the required distance every second. that is the interpretation of the formula for power. Notice that we don't have to use the concept of average velocity in this formula. in a constant velocity motion, the velocity will be same at any point we choose to measure.
We may also consider a zero friction and zero air drag scenario. There wont be any deceleration. Hence, we would not require any more power. We just need the power sufficient enough to take it to the required velocity and the bike will keep on moving. This is the condition is space. This is Galielo's law of inertia.
Stage 5: Now we need to decelerate. All that we need to do is to release the throttle completely and the bike will come to a stop. Friction and air drag will offer the required power. But if we have to stop it in a shorter distance and shorter time, we can apply brakes.
Having introduced the concept of external deceleration, we can modify the power calculation for stage 3.
Power = mass * (required acceleration + external deceleration) * average velocity.
So, in order to cause motion of constant velocity or constant acceleration all that we need is power. a continuous or varying power supply as long as (for the duration of time) we require motion.
Now it is clear why bike manufactures specify power (maximum power). Along with this they will also specify the mass and acceleration of the bike.
Going back to the formula p = m x a x d/t, we know p, m and a. we need to know either distance or time to calculate the remaining unknown. acceleration comes to our help again. a = (v-u)/t. v the final velocity, u the initial velocity (consider it as zero) and t the time taken for this change in velocity. from this we can find a. From the given power, we can now calculate how much displacement it will take to reach the required velocity. remember, the displacement in each second is not same. so we got to add the distance traveled in each second to arrive at total displacement.
Real life example 2: Lift
The bike example considered only the horizontal motion and we need not have to bother about vertical acceleration (it is important to note that a part of vertical acceleration would be used in calculating the amount of friction. for sake of simplicity, we just said friction is a decelerating action).
Now, a lift would involve motion is vertical upward and downward directions. so, what is the difference. the difference is that we have to move against gravity during the upward motion and move with (assisted by) gravity.
It might look complicated. but all that we need is to use the same logic that we used for bike.
Upward motion:
power = mass x (natural downward acceleration + upward acceleration required) x velocity.
Downward motion:
power = mass x (natural downward acceleration - downward acceleration required) x velocity.
You will observe that this is similar to horizontal motion with friction and air drag. Instead we have considered downward acceleration. an interesting point here is gravity will assist downward motion and oppose upward motion. but friction and air drag will always oppose motion.
It is good to introduce the concept of counter weight here. use of counter weight will reduce the power requirement and hence the size and cost of the motor can be reduced. How does a counter weight help?. simple. It will cancel out the natural acceleration.
For the sake of convenience we may add some more terms to our vocabulary by analyzing the concept of power.
varying power = m * a * average velocity -> instead of average use may use varying ever second.
continuous power = m * a * velocity
force = m * a
work = f x d. Work is another abstract concept measured in joules
Hence power = w/t or f * v depending on our interpretation and is measured in watts.
You might have also observed that we need not have to use the concept of force till now for solving real world problems. Force is yet another abstract concept. once we say force, we need to say across what distance and in what time interval and that becomes power.
Having understood intuitively, the concept of power, we shall get to the concept of work, energy (potential and kinetic) and also the concept of power of rest and power of collisions
Time Estimate for Reading: 30 min
Learning Objectives: Velocity, Accelration and Power
Effort Required: Medium
Pedagogy Model: Evolution, Formula Analysis, Inter-disciplinary
Prior Physics Concepts: displacement, velocity, acceleration, mass
Prior Math Tools: Secondary school level Arithmetic, geometry and algebra
We wish to analyse and predict the motion of objects. So far we have identified that the amount of motion can be measured using the concepts of displacement, velocity and acceleration. We also understood the significance of acceleration; the bridge between space, time and mass.
Just to refresh our memory, measurement of velocity requires two points. Measure the distance between them and divide by the time interval.
Similarly, measurement of acceleration requires 3 points. measure the velocity between points 1 and 2, 2 and 3. also clock the time interval between points 1 and 3. now acceleration can be calculated by taking the difference in velocities and dividing by the total time taken. By getting the 3 point closer and closer, we would get more accurate measurement of acceleration. Mathematically speaking we say acceleration at a point (we shall see how geometry helped newton to settle this issue). but in practice we need 3 points.
Now that we know how to measure, We know the effect. Having known the effect, we are interested in causing the effect. Let us try to understand the cause of motion; motion in a straight line with constant velocity or constant acceleration. For that we will have to add the concept of power to our vocabulary.
What is power?
Let us begin with the definition, take a real life example and try to understand it step by step.
Definition:
Power = mass of the object x required acceleration x velocity of the object. This might look incomprehensible in the beginning but it is rather a very simple concept. It would be all the more intriguing if you have come across the text book definition "power is rate of doing work or rate of spending energy". Similar to displacement, work and energy are abstract concepts. They cannot be experienced in real life. We wont need them for the time being. Let us defer a discussion on work and energy later. So all we need is power to cause motion.
Since we state that power is real, let us take a real life example of using a two wheeler. Say a bike. the bike may be a by cycle (mechanical power), gasoline power bike (thermal power) or an electric bike (Electric power). Notice that power can be provided from different sources.
Electric Power = Voltage x current
Thermal power = mass x Specific heat capacity x average temperature difference/second
Mechanical power (linear motion) = mass x acceleration x average velocity
Mechanical power (rotational motion) = 2 x pi x revs per second x mass x acceleration x radius
Do not bother if you have not understood the formulas. we will get to it shortly.
Real life example 1: Bike
State 1.Initially the bike is a rest (initial velocity is zero).
State 2: Mount on the bike and start the engine. Let us assume it has an automatic gear and the road is perfectly horizontal.
State 3:Turn the accelerator (throttle) handle and the bike starts moving. You experience a change in velocity as long as you keep turning the accelerator handle (within the limit of traffic and vehicle capability).
Note down the distance traveled and time taken to reach the required velocity
State 4: Once we reach the required velocity (we wont be using the word speed. speed is required only for academic purposes), notice that we will not be twisting the accelerator handle any more. may be if you have noticed carefully we will release it quite a bit once we reached the required velocity.
State 5: Once the destination is reached (we are considering only straight line motion for now), Notice the distance traveled and time taken at constant velocity. Now, we release the accelerator handle from current position, we will observe that the velocity automatically decreases. Again measure note the distance traveled and time taken to reach zero velocity. The rate of decreasing velocity is termed deceleration. Just the opposite of acceleration.
If acceleration or deceleration are not constant, we will observe jerk; and jerk is not a comfortable experience.
Let us now go on to calculate the power required at various states.
Consider mass of the bike (including the person) = m kg
State 3: Power should be equal to mass x required acceleration of the object x average velocity
therefore power required = m x a x d/t. and is measured in watts.
m x a is quite intuitive. but, we may ask why d/t?. The answer is, any displacement will take a certain amount of time. So to define power in a sentence, Power is a quantity which is a product of mass to be accelerated through space in a given amount of time. the given amount of time is one second. Observe that during an accelerated motion, the velocity will keep changing every second. So, we need to keep changing the power every second (reason for gradually twisting the throttle). we may say that we should provide a continuously varying power. notice that, if you happen to accelerate for a period of 2 secs, the power required in 1 sec is less that the one in the next second (velocity would have increased by now)
There are other simpler methods to calculate continuously varying power using abstract concepts like work and kinetic energy. the concept of kinetic energy had to wait for calculus and calculus was invented (though there are many others who have contributed) by newton and leibnitz. So, we shall defer it for now.
State 4: Now we need a constant velocity motion. We would be tempted to apply the same reasoning above and would arrive at a quantity m x v. But this is not so. Why?
We have friction and air drag acting in the opposite direction. They offer deceleration (the opposite of acceleration) and so we cannot use v. How to calculate friction and air drag will be covered in a separate article. for now, it is sufficient to know that they offer deceleration.
Now the formula for power = mass x external deceleration x velocity
Notice that the required acceleration of the object has become equal to external deceleration. We just need to keep accelerating through the required distance every second. that is the interpretation of the formula for power. Notice that we don't have to use the concept of average velocity in this formula. in a constant velocity motion, the velocity will be same at any point we choose to measure.
We may also consider a zero friction and zero air drag scenario. There wont be any deceleration. Hence, we would not require any more power. We just need the power sufficient enough to take it to the required velocity and the bike will keep on moving. This is the condition is space. This is Galielo's law of inertia.
Stage 5: Now we need to decelerate. All that we need to do is to release the throttle completely and the bike will come to a stop. Friction and air drag will offer the required power. But if we have to stop it in a shorter distance and shorter time, we can apply brakes.
Having introduced the concept of external deceleration, we can modify the power calculation for stage 3.
Power = mass * (required acceleration + external deceleration) * average velocity.
So, in order to cause motion of constant velocity or constant acceleration all that we need is power. a continuous or varying power supply as long as (for the duration of time) we require motion.
Now it is clear why bike manufactures specify power (maximum power). Along with this they will also specify the mass and acceleration of the bike.
Going back to the formula p = m x a x d/t, we know p, m and a. we need to know either distance or time to calculate the remaining unknown. acceleration comes to our help again. a = (v-u)/t. v the final velocity, u the initial velocity (consider it as zero) and t the time taken for this change in velocity. from this we can find a. From the given power, we can now calculate how much displacement it will take to reach the required velocity. remember, the displacement in each second is not same. so we got to add the distance traveled in each second to arrive at total displacement.
Real life example 2: Lift
The bike example considered only the horizontal motion and we need not have to bother about vertical acceleration (it is important to note that a part of vertical acceleration would be used in calculating the amount of friction. for sake of simplicity, we just said friction is a decelerating action).
Now, a lift would involve motion is vertical upward and downward directions. so, what is the difference. the difference is that we have to move against gravity during the upward motion and move with (assisted by) gravity.
It might look complicated. but all that we need is to use the same logic that we used for bike.
Upward motion:
power = mass x (natural downward acceleration + upward acceleration required) x velocity.
Downward motion:
power = mass x (natural downward acceleration - downward acceleration required) x velocity.
You will observe that this is similar to horizontal motion with friction and air drag. Instead we have considered downward acceleration. an interesting point here is gravity will assist downward motion and oppose upward motion. but friction and air drag will always oppose motion.
It is good to introduce the concept of counter weight here. use of counter weight will reduce the power requirement and hence the size and cost of the motor can be reduced. How does a counter weight help?. simple. It will cancel out the natural acceleration.
For the sake of convenience we may add some more terms to our vocabulary by analyzing the concept of power.
varying power = m * a * average velocity -> instead of average use may use varying ever second.
continuous power = m * a * velocity
force = m * a
work = f x d. Work is another abstract concept measured in joules
Hence power = w/t or f * v depending on our interpretation and is measured in watts.
You might have also observed that we need not have to use the concept of force till now for solving real world problems. Force is yet another abstract concept. once we say force, we need to say across what distance and in what time interval and that becomes power.
Having understood intuitively, the concept of power, we shall get to the concept of work, energy (potential and kinetic) and also the concept of power of rest and power of collisions
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