Position Velocity Acceleration vectors - Parabolic motion

Problem Statement:

A skier jumps from the top of a cliff of height h = 30 m. His initial velocity has a magnitude of 40 m/s and direction α = 300 (see figure).

1. Calculate the position, velocity and acceleration vectors of the skier as a function of time.
2. Calculate his time of flight.
3. Calculate his distance from the cliff at this moment.
4. Calculate his velocity at this moment.

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Solution:

We are going to calculate the position, velocity and acceleration vectors with respect to O using the Cartesian axes depicted in the figure.

The acceleration vector is given by:

This vector points in the positive direction of the y axis.

The initial velocity vector is given by:

And substituting the givens:

If you want to see in more detail how these components are calculated check out Problem 6.

To calculate the velocity vector of the skier as a function of time we use the definition of the accelation vector and integrate:

In Problem 1 you can see in more detail how this integral is solved.

Substituting the initial velocity and grouping terms:

Next we are going to calculate the position vector of the skier. His initial position vector is zero because he jumps from the origin of coordinates. Integrating the definition of the velocity vector:

And substituting the velocity vector:

We are going to write each component of this vector for later use:

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When the skier reaches the ground the y component of his position vector will be y = h = 30 m. Therefore we are going to substitute this condition in equation (2):

The flight time will be the positive root of the polynomial.

We substitute this time into equation (1) to calculate his distance from the cliff at this moment:

We substitute the same time into the velocity vector:

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