The work W0 accelerates a car from 0 to 20 km/h. How much work (in terms of W0) is required to accelerate the car from 20 km/h to 100 km/h?
Let's assume the acceleration, a , is constant
then the force is constant
F = m a
work = integral of force dx from low speed to high speed
which is force * (Xend - Xbegin) because F is constant
so we need to know x begin and x end
let x begin be 0
then
d^2x/dt^2 = a
dx/dt = v = Vi + a t
20 = 0 + a t
so t = 20/a
x = 0 + Vi t + (1/2) a t^2
Xend = (1/2) a (400/a^2) = 200/a
so
Wo = m a ( 200/a) = 200 m
now the second part
integrate the same thing but Vi = 20 not 0
F = m a still so
W = m a [ Xend - Xbegin] again
let x begin = 0 again
Vi = 20
v = 20 + a t
40 = 20 + a t
t = 20/a same time for the second part
but now our average speed is 60
Xend = 1200/a
so
W = m a (1200/a) = m (1200)
W/Wo = 1200/200 = 6
W = 6 Wo
by the way that means we need 6 times the power because the time is the same
To find the work required to accelerate the car from 20 km/h to 100 km/h, we can use the work-energy principle.
The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. In this case, we can assume that the car is moving in a straight line, and there are no other forces acting on it apart from the work done to accelerate it.
The kinetic energy of an object can be calculated using the equation:
KE = (1/2) * m * v^2
where KE is the kinetic energy, m is the mass of the object, and v is its velocity.
Now, to find the work done to accelerate the car from 20 km/h to 100 km/h, we need to calculate the change in kinetic energy. The initial kinetic energy is given by:
KE_initial = (1/2) * m * v_initial^2
where v_initial is the initial velocity (20 km/h) and m is the mass of the car.
Similarly, the final kinetic energy is given by:
KE_final = (1/2) * m * v_final^2
where v_final is the final velocity (100 km/h).
The change in kinetic energy, ΔKE, is then given by:
ΔKE = KE_final - KE_initial
Now, let's express the velocities in terms of m/s since the SI unit system is commonly used in physics.
To convert km/h to m/s, remember that 1 km/h is equal to (1/3.6) m/s.
So, the initial velocity in m/s is:
v_initial = 20 km/h * (1/3.6) m/s
And the final velocity in m/s is:
v_final = 100 km/h * (1/3.6) m/s
Substituting these values back into the equation for the change in kinetic energy:
ΔKE = (1/2) * m * (v_final^2 - v_initial^2)
Now, we can compare this work W1 to the original work W0 required to accelerate the car from 0 to 20 km/h.
The work done to accelerate the car from 20 km/h to 100 km/h is equal to the change in kinetic energy:
W1 = ΔKE
So, the work required to accelerate the car from 20 km/h to 100 km/h can be calculated by substituting the values we have into the formula:
W1 = (1/2) * m * (v_final^2 - v_initial^2)
where v_final is the final velocity in m/s (100 km/h * (1/3.6) m/s) and v_initial is the initial velocity in m/s (20 km/h * (1/3.6) m/s).
I hope this explanation helps!