The triple jump is a track-and-field event in which an...

The triple jump is a track-and-field event in which an athlete gets a running start and tries to leap as far as he can with a hop, step, and jump. Shown in the figure is the initial hop of the athlete. Assuming that he approaches the takeoff line from the left with a horizontal velocity of $10 \mathrm{~m} / \mathrm{s}$, remains in contact with the ground for $0.18 \mathrm{~s}$, and takes off at a $50^{\circ}$ angle with a velocity of $12 \mathrm{~m} / \mathrm{s},$ determine the vertical component of the average impulsive force exerted by the ground on his foot. Give your answer in terms of the weight $W$ of the athlete.

Key Concepts

Gravitational Force and Its Impulse

The weight of an object represents the gravitational force acting on it. In dynamics problems involving upward jumps or lifts, gravity exerts a constant force that contributes an impulse (force multiplied by the time interval) over the duration of contact, which must be accounted for in the net change of momentum.

Vector Decomposition of Velocity

When analyzing the motion of projectiles or jumps, the initial velocity is often broken down into its horizontal and vertical components. This decomposition simplifies the study of separate motions in perpendicular directions, which is especially helpful when considering forces acting in only one of those directions.

Conversion to Weight Terms

Often, forces are expressed in terms of the object’s weight to provide a convenient, dimensionless comparison. This requires recognizing that weight is the product of mass and the acceleration due to gravity, thus linking the dynamics of the motion directly to the gravitational pull on the object.

Impulse–Momentum Theorem

This theorem relates the impulse applied to an object (the product of force and the time interval over which it acts) to the change in the object's momentum. It is crucial in analyzing situations where forces act briefly to change the motion of an object, such as in the takeoff phase of the triple jump.

Average Force Over a Time Interval

In many problems involving impulsive forces, an average force is defined by dividing the total impulse by the time over which the force is applied. This concept is used to determine an effective or average force when dealing with non-continuous or highly concentrated forces during short contact times.

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