The drawing shows a skateboarder moving at 5.4 m / s along a horizontal section of a track that

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The drawing shows a skateboarder moving at 5.4 m / s along a...

The drawing shows a skateboarder moving at $5.4 \mathrm{m} / \mathrm{s}$ along a horizontal section of a track that is slanted upward by $48^{\circ}$ above the horizontal at its end, which is $0.40 \mathrm{m}$ above the ground. When she leaves the track, she follows the characteristic path of projectile motion. Ignoring friction and air resistance, find the maximum height $H$ to which she rises above the end of the track.

Key Concepts

Trigonometric Functions

Trigonometric functions such as sine and cosine are crucial when resolving the initial velocity vector into vertical and horizontal components. This resolution allows for separate analysis of the effects of gravity on the vertical component while the horizontal component remains unchanged, facilitating the calculation of variables like maximum height in projectile motion.

Kinematic Equations for Vertical Motion

The kinematic equations describe the motion of objects under constant acceleration. For vertical motion, the maximum height reached above the launch point can be determined using the equation that relates the vertical component of the velocity, gravitational acceleration, and displacement. Specifically, the equation (v_y)^2/(2g) gives the additional height attained before the vertical speed becomes zero at the peak of the trajectory.

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and moves under the influence of gravity only, following a curved path. This concept involves analyzing the motion separately in the horizontal and vertical directions since the horizontal component of velocity remains constant (ignoring air resistance) while the vertical component is affected by gravity.

Decomposition of Velocity

When an object is launched at an angle, its initial velocity has both horizontal and vertical components. Decomposing the velocity using trigonometric functions (sine for the vertical component and cosine for the horizontal component) is essential to analyze the motion, especially to determine parameters like the maximum height reached in the vertical motion.

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