The driver of an 800.0 kg car decides to double the speed from 25.0 m/s to 50.0 m/s. What effect would this have on the amount of work required to stop the car, that is, on the kinetic energy of the car?
KE grows by a factor of 4.
The driver of a 850.0 kg car decides to double the speed from 23.2 m/s to 46.4 m/s. What effect would this have on the amount of work required to stop the car, that is, on the kinetic energy of the car?
To analyze the effect on the amount of work required to stop the car, we need to examine the relationship between kinetic energy and speed.
The kinetic energy of an object is given by the equation:
KE = (1/2) * m * v^2
Where:
KE = Kinetic Energy
m = mass of the object
v = velocity (speed) of the object
In this case, the mass of the car (m) is given as 800.0 kg. Initially, the speed of the car (v_initial) is 25.0 m/s, and after doubling the speed, the final speed (v_final) becomes 50.0 m/s.
Let's now calculate the initial and final kinetic energies of the car:
Initial Kinetic Energy (KE_initial):
KE_initial = (1/2) * m * v_initial^2
= (1/2) * 800.0 kg * (25.0 m/s)^2
= 0.5 * 800.0 kg * 625.0 m^2/s^2
= 250,000 Joules
Final Kinetic Energy (KE_final):
KE_final = (1/2) * m * v_final^2
= (1/2) * 800.0 kg * (50.0 m/s)^2
= 0.5 * 800.0 kg * 2500.0 m^2/s^2
= 2,000,000 Joules
The initial kinetic energy of the car is 250,000 Joules, while the final kinetic energy is 2,000,000 Joules. Thus, doubling the speed leads to an increase in the amount of work required to stop the car, which is equivalent to an increase in the kinetic energy of the car.
To determine the effect of doubling the speed on the amount of work required to stop the car, we need to analyze the kinetic energy of the car.
The kinetic energy of an object can be calculated using the formula:
Kinetic Energy = (1/2) * Mass * Velocity^2
Given that the mass of the car is 800.0 kg, and the initial speed is 25.0 m/s, we can calculate the initial kinetic energy:
Initial Kinetic Energy = (1/2) * 800.0 kg * (25.0 m/s)^2
Now, let's calculate the kinetic energy when the speed is doubled to 50.0 m/s:
Final Kinetic Energy = (1/2) * 800.0 kg * (50.0 m/s)^2
To compare the initial and final kinetic energies, let's calculate the ratio of the two:
Ratio = Final Kinetic Energy / Initial Kinetic Energy
By substituting the values into the formula, we get:
Ratio = [(1/2) * 800.0 kg * (50.0 m/s)^2] / [(1/2) * 800.0 kg * (25.0 m/s)^2]
Simplifying the equation, we see that the mass of the car cancels out:
Ratio = (50.0 m/s)^2 / (25.0 m/s)^2
Squaring the values gives us:
Ratio = 2500 m^2/s^2 / 625 m^2/s^2
Simplifying further, we have:
Ratio = 4
Therefore, doubling the speed of the car results in the kinetic energy being four times greater. In other words, the amount of work required to stop the car quadruples when the speed is doubled.