A 5 kg ball rolls off a 34 m high cliff, and lands 29 m from the base of the cliff. Express the displacement and the gravitational force in terms of vectors and calculate the work done by the gravitational force. Note that the gravitational force is < 0, -mg, 0 >, where g is a positive number (+9.8 N/kg).

To solve this problem, we can break it down into a few steps:

Step 1: Displacement

To find the displacement vector, we need to subtract the initial position vector from the final position vector. The ball rolls off a 34 m high cliff, so the initial position vector is <0, 0, 34>. It lands 29 m from the base of the cliff, so the final position vector is <29, 0, 0>. Subtracting the initial position from the final position gives us:

Displacement vector = Final position vector - Initial position vector
= <29, 0, 0> - <0, 0, 34>
= <29, 0, -34>

Therefore, the displacement vector is <29, 0, -34>.

Step 2: Gravitational Force

The gravitational force is given as <0, -mg, 0>, where m is the mass of the ball and g is the acceleration due to gravity. The mass of the ball is 5 kg, and the acceleration due to gravity is 9.8 m/s². Multiplying the mass by the acceleration due to gravity gives us:

Gravitational force = -mg
= -5 * 9.8
= -49 N

Therefore, the gravitational force is -49 N.

Step 3: Work Done

To calculate the work done by the gravitational force, we need to multiply the force vector by the displacement vector.

Work done = Gravitational force * Displacement
= (-49) * <29, 0, -34>

Multiplying the components of the vectors gives us:

Work done = (-49 * 29) + (0 * 0) + (-49 * (-34))
= -1421 + 0 + 1666
= 245 N*m

Therefore, the work done by the gravitational force is 245 N*m.