Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Let's tackle question 57 step-by-step:
To find the minimum time required for a vehicle of mass [tex]\( m \)[/tex] being driven by an engine of power [tex]\( P \)[/tex] to accelerate from rest, we analyze the kinematics and the relationship between power, force, and acceleration.
1. Understand Power and Force Relationships:
- Power ([tex]\( P \)[/tex]) is the rate at which work is done.
- Work done ([tex]\( W \)[/tex]) is given by [tex]\( W = F \cdot d \)[/tex], where [tex]\( F \)[/tex] is the force and [tex]\( d \)[/tex] is the distance.
- We know that power is also given by the product of force and velocity: [tex]\( P = F \cdot v \)[/tex].
2. Relate Force and Acceleration:
- Newton's second law states [tex]\( F = m \cdot a \)[/tex], where [tex]\( a \)[/tex] is the acceleration.
- Substituting [tex]\( F = m \cdot a \)[/tex] in the power equation [tex]\( P = F \cdot v \)[/tex], we get:
[tex]\[ P = m \cdot a \cdot v. \][/tex]
- Rearranging for acceleration, [tex]\( a \)[/tex]:
[tex]\[ a = \frac{P}{m \cdot v}. \][/tex]
3. Set Up the Differential Equation:
- The acceleration [tex]\( a \)[/tex] is the time derivative of velocity ([tex]\( v \)[/tex]):
[tex]\[ a = \frac{dv}{dt}. \][/tex]
- Thus:
[tex]\[ \frac{dv}{dt} = \frac{P}{m \cdot v}. \][/tex]
- Rearrange and separate variables:
[tex]\[ v \, dv = \frac{P}{m} \, dt. \][/tex]
4. Integrate to Find Time:
- Integrate both sides to find the velocity as a function of time:
[tex]\[ \int_{0}^{v} v \, dv = \frac{P}{m} \int_{0}^{t} dt. \][/tex]
- The integrals yield:
[tex]\[ \left[ \frac{v^2}{2} \right]_{0}^{v} = \frac{P}{m} \left[ t \right]_{0}^{t}. \][/tex]
- Simplify the result:
[tex]\[ \frac{v^2}{2} = \frac{P}{m} t. \][/tex]
- Solving for time [tex]\( t \)[/tex]:
[tex]\[ t = \frac{m v^2}{2P}. \][/tex]
Therefore, the minimum time required for a vehicle of mass [tex]\( m \)[/tex] driven by an engine of power [tex]\( P \)[/tex] to reach a velocity [tex]\( v \)[/tex] from rest is:
[tex]\[ t = \frac{m v^2}{2P}. \][/tex]
To find the minimum time required for a vehicle of mass [tex]\( m \)[/tex] being driven by an engine of power [tex]\( P \)[/tex] to accelerate from rest, we analyze the kinematics and the relationship between power, force, and acceleration.
1. Understand Power and Force Relationships:
- Power ([tex]\( P \)[/tex]) is the rate at which work is done.
- Work done ([tex]\( W \)[/tex]) is given by [tex]\( W = F \cdot d \)[/tex], where [tex]\( F \)[/tex] is the force and [tex]\( d \)[/tex] is the distance.
- We know that power is also given by the product of force and velocity: [tex]\( P = F \cdot v \)[/tex].
2. Relate Force and Acceleration:
- Newton's second law states [tex]\( F = m \cdot a \)[/tex], where [tex]\( a \)[/tex] is the acceleration.
- Substituting [tex]\( F = m \cdot a \)[/tex] in the power equation [tex]\( P = F \cdot v \)[/tex], we get:
[tex]\[ P = m \cdot a \cdot v. \][/tex]
- Rearranging for acceleration, [tex]\( a \)[/tex]:
[tex]\[ a = \frac{P}{m \cdot v}. \][/tex]
3. Set Up the Differential Equation:
- The acceleration [tex]\( a \)[/tex] is the time derivative of velocity ([tex]\( v \)[/tex]):
[tex]\[ a = \frac{dv}{dt}. \][/tex]
- Thus:
[tex]\[ \frac{dv}{dt} = \frac{P}{m \cdot v}. \][/tex]
- Rearrange and separate variables:
[tex]\[ v \, dv = \frac{P}{m} \, dt. \][/tex]
4. Integrate to Find Time:
- Integrate both sides to find the velocity as a function of time:
[tex]\[ \int_{0}^{v} v \, dv = \frac{P}{m} \int_{0}^{t} dt. \][/tex]
- The integrals yield:
[tex]\[ \left[ \frac{v^2}{2} \right]_{0}^{v} = \frac{P}{m} \left[ t \right]_{0}^{t}. \][/tex]
- Simplify the result:
[tex]\[ \frac{v^2}{2} = \frac{P}{m} t. \][/tex]
- Solving for time [tex]\( t \)[/tex]:
[tex]\[ t = \frac{m v^2}{2P}. \][/tex]
Therefore, the minimum time required for a vehicle of mass [tex]\( m \)[/tex] driven by an engine of power [tex]\( P \)[/tex] to reach a velocity [tex]\( v \)[/tex] from rest is:
[tex]\[ t = \frac{m v^2}{2P}. \][/tex]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.