An older-model car accelerates from 0 to speed v in 10. s. A newer, more powerful sports car of the same mass accelerates from 0 to 4v in the same time period. Assuming the energy coming from the engine appears only as kinetic energy of the cars, compare the power of the two cars. (Take Pnew and Pold as the powers of the new car and old car, respectively.)
Pnew = Pold
Let's solve this step by step.
Step 1: Find the kinetic energy of the older-model car.
The kinetic energy (KE) of a car is given by the equation KE = (1/2)mv^2, where m is the mass of the car and v is the velocity.
Step 2: Find the kinetic energy of the newer sports car.
Since the newer car accelerates to a speed of 4v, we can substitute 4v for v in the kinetic energy equation: KEnew = (1/2)m(4v)^2.
Step 3: Use the definition of power to find the power of each car.
Power (P) is defined as the rate at which work is done or energy is transferred. It is given by the equation P = ΔE/Δt, where ΔE is the change in energy and Δt is the change in time.
Step 4: Compare the powers of the two cars.
Based on the given information, we can compare the powers by comparing the ratio of the kinetic energies of the cars in the same time period.
Now, let's carry out the calculations.
Step 1: Calculate the kinetic energy of the older-model car.
KEold = (1/2)m(velocity)^2.
Step 2: Calculate the kinetic energy of the newer sports car.
KEnew = (1/2)m(4v)^2 = 8(1/2)mv^2 = 8KEold.
Step 3: Calculate the power of each car.
Pold = ΔEold/Δt = KEold/Δt.
Pnew = ΔEnew/Δt = KEnew/Δt.
Step 4: Compare the powers of the two cars.
Pnew = 8Pold.
Therefore, the power of the newer sports car is 8 times greater than the power of the older-model car.
To compare the power of the two cars, we need to calculate the power for each car and compare them.
Power is defined as the rate at which work is done or the rate at which energy is transferred. It can be calculated using the formula:
Power = Work / Time
In this case, the energy coming from the engine appears only as kinetic energy of the cars. Therefore, the work done by the engine is equal to the change in kinetic energy of the cars.
For the older-model car:
The initial kinetic energy is 0, and the final kinetic energy is (1/2)m v^2, where m is the mass of the car and v is the final speed.
So, the work done by the engine is (1/2)m v^2.
Similarly, for the newer sports car:
The initial kinetic energy is 0, and the final kinetic energy is (1/2)m (4v)^2, since the car accelerates to 4 times the speed.
So, the work done by the engine is (1/2)m (4v)^2.
Now, we need to divide the work done by the time taken (which is 10s) for both cars to calculate their respective powers.
For the older-model car:
Power_old = (1/2)m v^2 / 10s
For the newer sports car:
Power_new = (1/2)m (4v)^2 / 10s
Now we can compare the two powers:
Pnew / Pold = ((1/2)m (4v)^2 / 10s) / ((1/2)m v^2 / 10s)
By canceling out the common factors, we get:
Pnew / Pold = (4v)^2 / v^2
Simplifying further:
Pnew / Pold = 16
So, the power of the new sports car is 16 times the power of the older-model car.
power = work/time
since the time is the same, the power ratio is the work ratio
work in = kinetic energy out
kinetic energy proportional to v^2 [ like (1/2)m v^2 ]
v'^2/v^2 = 16