To calculate the gain in kinetic energy of the lawn mower after 10 seconds, we need to consider the work done on the mower and the energy dissipated due to friction.
First, let's calculate the work done by the person pushing the lawn mower. The work done is given by the formula:
Work = Force * Distance * cos(θ)
where:
- Force is the applied force (15 N),
- Distance is the displacement of the lawn mower,
- θ is the angle between the applied force and the direction of motion (45 degrees).
Since the lawn mower is pushed in the direction of the applied force, the angle θ is 0 degrees. Therefore, cos(θ) = cos(0) = 1.
Work = 15 N * Distance * cos(0) = 15 N * Distance
Next, let's calculate the distance covered by the lawn mower. We can use the kinematic equation:
Distance = (1/2) * acceleration * time^2
The acceleration of the lawn mower can be calculated using Newton's second law:
Force - Frictional force = mass * acceleration
The frictional force can be calculated using the equation:
Frictional force = coefficient of friction * normal force
The normal force is equal to the weight of the lawn mower, which is given by:
Normal force = mass * gravitational acceleration
Substituting these values into the equations:
Normal force = 6 kg * 9.8 m/s^2 = 58.8 N
Frictional force = 0.1 * 58.8 N = 5.88 N
Force - Frictional force = mass * acceleration
15 N - 5.88 N = 6 kg * acceleration
9.12 N = 6 kg * acceleration
acceleration = 9.12 N / 6 kg ≈ 1.52 m/s^2
Distance = (1/2) * 1.52 m/s^2 * (10 s)^2 = 76 m
Now that we have the distance covered, we can calculate the work done:
Work = 15 N * 76 m = 1140 J
The gain in kinetic energy is equal to the work done. Therefore, the gain in kinetic energy of the lawn mower after 10 seconds is 1140 J.