..A lawn roller is pushed across a lawn by a force of 103 N along the direction of the handle, which is 22.5° above the horizontal. If 63.1 W of power is developed for 90.0 s, what distance is the roller pushed?

To find the distance the roller is pushed, we can use the formula for power:

Power = Force x Velocity

In this case, we have the power (63.1 W) and the time (90.0 s), but we need to find the velocity. We can use the equation of motion to find the velocity:

Velocity = Distance / Time

Since the roller is pushed along the direction of the handle, which is at a 22.5° angle above the horizontal, we can decompose the force into two components: one in the horizontal direction and one in the vertical direction. The component in the horizontal direction is given by:

Horizontal Force = Force x cos(angle)

And the component in the vertical direction is given by:

Vertical Force = Force x sin(angle)

In this case, the angle is 22.5°, so we can substitute the values:

Horizontal Force = 103 N x cos(22.5°)
Vertical Force = 103 N x sin(22.5°)

Now, we can find the acceleration in the horizontal direction using Newton's second law:

F = m x a

Where F is the force in the horizontal direction, m is the mass of the roller, and a is the acceleration. Rearranging the equation, we get:

a = F / m

Now, we can find the velocity using another equation of motion:

Velocity = Initial Velocity + (Acceleration x Time)

In this case, the initial velocity is assumed to be zero since the roller starts from rest. So the equation becomes:

Velocity = (Acceleration x Time)

Now, we can substitute the values to find the velocity:

Velocity = (Horizontal Force / m) x Time

Finally, we can substitute the velocity into the formula for power to get:

63.1 W = Force x Velocity

Then, we solve for the force:

Force = 63.1 W / Velocity

Now, we can substitute the force into the equation for distance:

Distance = Force x Time

Now, we can substitute the values to find the distance:

Distance = (63.1 W / Velocity) x Time

Finally, we need to convert the distance from meters to centimeters or any other unit.