Matthew is waterskiing. As the boat starts moving, he is at an angle of 8.0° to the right of the boat. The boat applies 250 newtons over the first 50 meters, while the angle stays at roughly 8.0°. What work has the boat done on Matthew?
To find the work done by the boat on Matthew, we can use the formula for work:
Work = Force x Displacement x cos(theta)
where:
- Force is the applied force by the boat,
- Displacement is the distance covered by Matthew,
- theta is the angle between the applied force and the displacement.
Given:
- Force = 250 Newtons,
- Displacement = 50 meters,
- theta = 8.0°.
We need to convert the angle from degrees to radians in order to use the formula. There are 180° in π radians, so:
theta_radians = (8.0° * π) / 180°
theta_radians ≈ 0.1396 radians.
Now we can calculate the work done by the boat on Matthew:
Work = 250 N * 50 m * cos(0.1396 radians)
Calculating cos(0.1396 radians) ≈ 0.9904.
Work ≈ 250 N * 50 m * 0.9904
Work ≈ 12,510 Joules
Therefore, the boat has done approximately 12,510 Joules of work on Matthew.
To find the work done by the boat on Matthew, we need to calculate the dot product of the force vector and the displacement vector. The formula for work is:
Work = Force * Displacement * cos(angle)
In this case, the force applied by the boat is 250 Newtons. The displacement is 50 meters. And the angle is roughly 8.0°.
However, before we can calculate the work, we need to convert the angle from degrees to radians, since the cosine function in the formula requires the angle to be in radians.
To convert degrees to radians, we use the formula:
Radians = Degrees * π / 180
Let's convert the angle:
Angle in radians = 8.0° * π / 180 = 0.139626 radians (approximately)
Now we can calculate the work:
Work = 250 Newtons * 50 meters * cos(0.139626 radians)
To calculate the cosine of the angle, you can use a calculator or a mathematical software.
After substituting the values, you can multiply the numbers to get the work done by the boat on Matthew.
250 cos 8 N which is force in direction of motion
times distance in that direction which is 50 m
= 12,378 Joules