A stopped 100okg car exerts 160kW of power as it accelerate. If it is only 75% efficient, how far will it travel in 4 seconds?
To calculate the distance traveled by the car in 4 seconds, we need to follow these steps:
Step 1: Calculate the force exerted by the car.
The power exerted by the car is given as 160 kW. Power is defined as the rate at which work is done or the rate at which energy is transferred. Since power is the product of force and velocity, we can rearrange this equation to solve for force:
Power = Force × Velocity
In this case, we know the power is 160 kW and we need to find the force. However, we don't have the velocity. Therefore, we need to determine the velocity of the car first.
Step 2: Find the velocity of the car.
To find the velocity, we need to know the amount of time the car accelerates for.
Given:
Mass of the car (m) = 1000 kg
Power (P) = 160 kW
Time (t) = 4 s
We know that power is the work done or energy transferred per unit time. In this case, we can use the equation:
Power = Work/Time
Combining the equation for work (Work = Force × Distance) with the equation for power, we can relate force, distance, and time:
Force × Distance/Time = Power
Force × Distance = Power × Time
Force = (Power × Time)/Distance
Since we have the power and time, we need to determine the distance traveled in order to calculate the force. We can use Newton's second law of motion to determine the distance:
Force = Mass × Acceleration
Acceleration = Change in Velocity/Time
In this case, the acceleration is unknown, but we know the initial velocity is 0 m/s and the time is 4 s. Therefore, the final velocity (v) can be determined using the formula:
v = u + at
Where:
v = final velocity
u = initial velocity (0 m/s)
a = acceleration
t = time (4 seconds)
Considering the car was initially at rest, the equation simplifies to:
v = at
Rearranging the equation, we can solve for acceleration of the car:
a = v/t
a = v/4
Finally, we substitute the calculated value of acceleration into Newton's second law of motion:
Force = Mass × Acceleration
Force = 1000 kg × (v/4)
Step 3: Calculate the distance traveled.
Now that we have determined the force, we can calculate the distance traveled by rearranging Newton's second law of motion again:
Distance = (Force × Time^2)/(2 × Mass)
Given:
Force = 1000 kg × (v/4)
Time = 4 s
Mass = 1000 kg
Substituting the values, we get:
Distance = (1000 kg × (v/4) × (4 s)^2)/(2 × 1000 kg)
Distance = v × (4 s)/2
Distance = 2v
So, the distance traveled is equal to twice the final velocity of the car after 4 seconds.