A 2.0 x 10^3 kg car accelerates from rest under the actions of two forces. One is a forward force of 1140N provided by traction between the wheels and the road. The other is a 950N resistive force due to various frictional forces. Use the work - kinetic energy theorem to determine how far the car must travel for its speed to reach 2.0m/s
The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. Mathematically, it can be expressed as:
Work = ΔKinetic Energy
The net work done on the car is equal to the sum of the work done by the traction force and the work done by the resistive force.
Net Work = Work by Traction Force + Work by Resistive Force
The equation for work is given by:
Work = Force x Distance x cosθ
where θ is the angle between the force and the direction of motion.
Since the car starts from rest, its initial kinetic energy is zero. The final kinetic energy is given by:
Final Kinetic Energy = 0.5 x mass x final velocity^2
Plugging in the given values, we have:
Final Kinetic Energy = 0.5 x 2000 kg x (2 m/s)^2
= 4000 J
Now, let's calculate the work done by the traction force. The traction force is a forward force, so the angle θ between the force and the direction of motion is 0°.
Work by Traction Force = 1140 N x Distance x cos0°
= 1140 N x Distance
Similarly, let's calculate the work done by the resistive force. The resistive force is acting opposite to the direction of motion, so the angle θ between the force and the direction of motion is 180°.
Work by Resistive Force = 950 N x Distance x cos180°
= -950 N x Distance
Now, let's substitute these values into the net work equation and solve for the distance:
Net Work = Work by Traction Force + Work by Resistive Force
4000 J = 1140 N x Distance - 950 N x Distance
4000 J = (1140 N - 950 N) x Distance
4000 J = 190 N x Distance
Distance = 4000 J / 190 N
Distance ≈ 21.05 m
Therefore, the car must travel approximately 21.05 meters for its speed to reach 2.0 m/s.