A 1000kg car accelerates from rest to 30m/s over a distance of 20m.
a) ignoring friction and air resistance, what is the power generated by the motor?
b) if the friction and air resistance is 200N, what is the power generated by the motor?
a) Power = (Kinetic energy increase)/(time)
Time*Average speed = t*Vfinal/2 = 20 m
Time = 1.333 seconds
Power = (1/2) M Vfinal^2/1.333= 3.376*10^5 J
= 453 Hp
b) Add 200N*20m = 4000J to the KE increase, and divide by 1.333 s
The new answer is 3.406*10^5 J = 457 Hp
To find the power generated by the motor in both cases, we need to use the work-energy principle and calculate the work done on the car.
a) ignoring friction and air resistance:
The work done on an object is equal to the change in its kinetic energy. In this case, the car accelerates from rest to 30 m/s, so the change in kinetic energy is given by:
ΔKE = 0.5 * m * (v_f^2 - v_i^2)
where ΔKE is the change in kinetic energy, m is the mass of the car, v_i is the initial velocity (0 m/s), and v_f is the final velocity (30 m/s).
Substituting the given values:
ΔKE = 0.5 * 1000 kg * (30 m/s)^2
Next, we need to calculate the time it takes the car to accelerate. We can use the equation of motion:
v_f = v_i + a * t,
where a is the acceleration and t is the time. Rearranging the equation, we get:
t = (v_f - v_i) / a.
Here, v_f = 30 m/s, v_i = 0 m/s, and we need to find the acceleration. To calculate the acceleration, we can use the equation:
a = Δv / Δt,
where Δv is the change in velocity and Δt is the change in time. In this case, Δv is the final velocity (30 m/s), and Δt is the distance traveled (20 m) divided by the average velocity:
Δt = d / v_avg = 20 m / ((v_i + v_f) / 2).
Substituting the given values:
Δt = 20 m / ((0 m/s + 30 m/s) / 2).
Now that we have Δv and Δt, we can calculate the acceleration:
a = Δv / Δt = 30 m/s / (20 m / ((0 m/s + 30 m/s) / 2)).
Finally, we can substitute the calculated values of acceleration and mass into the equation for work:
Work = ΔKE = 0.5 * 1000 kg * (30 m/s)^2.
The power generated by the motor is equal to the work done divided by the time taken:
Power = Work / t.
Calculate both the work and power using the respective formulas, and you will have the answer to part (a).
b) if the friction and air resistance is 200N:
In this case, we need to consider the external force acting on the car due to friction and air resistance. The net force acting on the car is equal to the sum of the external forces:
F_net = F_motor - F_friction.
Since F_friction is given as 200 N, we need to calculate F_motor. The net force can also be calculated using Newton's second law:
F_net = m * a,
where m is the mass and a is the acceleration. Rearranging the equation, we can find the acceleration:
a = F_net / m.
Using the value of F_net calculated from the sum of F_motor and F_friction, and the given mass of the car, we can find the acceleration:
a = (F_motor - F_friction) / m.
Once we have the acceleration, we can repeat the steps from part (a) to find the work done and then calculate the power generated by the motor.
Remember to adjust the equation for work by subtracting the work done against friction:
Work = (0.5 * m * (v_f^2 - v_i^2)) - (F_friction * d).
Finally, calculate the power using the formula: Power = Work / t.
By following these steps, you will be able to find the power generated by the motor in both cases, considering and ignoring friction and air resistance.