A car of mass 600 kg reduces speed from 90 km/h to 54 km/h in 15 seconds. Determine the braking power required to give this change of speed.
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A car of mass 600kg reduced speed from 90km\h to 54km\h in 5s .determine the braking power required to give this change of speed
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To determine the braking power required, we need to calculate the work done by the brakes in slowing down the car. The work is equal to the change in kinetic energy.
The initial kinetic energy (K1) is given by the formula:
K1 = (1/2) * m * v1^2
where:
m = mass of the car
v1 = initial velocity
The final kinetic energy (K2) is given by the formula:
K2 = (1/2) * m * v2^2
where:
v2 = final velocity
The change in kinetic energy (∆K) is given by the difference between K2 and K1:
∆K = K2 - K1 = (1/2) * m * v2^2 - (1/2) * m * v1^2
Now, we can calculate the values to substitute into the formula.
Given:
m = 600 kg (mass of the car)
v1 = 90 km/h (initial velocity)
v2 = 54 km/h (final velocity)
First, we need to convert the velocities from km/h to m/s, as the formula requires velocities in meters per second.
Converting v1 from km/h to m/s:
v1 = 90 km/h * (1000 m/1 km) * (1 h/3600 s)
v1 = 25 m/s
Converting v2 from km/h to m/s:
v2 = 54 km/h * (1000 m/1 km) * (1 h/3600 s)
v2 = 15 m/s
Now, we can substitute the values into the formula to calculate the change in kinetic energy (∆K):
∆K = (1/2) * m * v2^2 - (1/2) * m * v1^2
∆K = (1/2) * 600 kg * (15 m/s)^2 - (1/2) * 600 kg * (25 m/s)^2
∆K = (1/2) * 600 kg * 225 m^2/s^2 - (1/2) * 600 kg * 625 m^2/s^2
Now, we can simplify the equation:
∆K = (1/2) * 135000 kg * m^2/s^2 - (1/2) * 375000 kg * m^2/s^2
∆K = 67500 kg * m^2/s^2 - 187500 kg * m^2/s^2
∆K = -120000 kg * m^2/s^2
The negative sign indicates that work is done on the car (braking) rather than by the car.
Finally, we can determine the braking power using the formula:
Power = ∆K / t
Given:
t = 15 seconds (time taken)
Power = -120000 kg * m^2/s^2 / 15 s
Power = -8000 kg * m^2/s^3
The braking power required to give this change of speed is -8000 kg * m^2/s^3 (in the opposite direction of the car's motion).