A firefighter 50.0 m away from a burning building directs a...

A firefighter $50.0 \mathrm{~m}$ away from a burning building directs a stream of water from a fire hose at an angle of $30.0^{\circ}$ above the horizontal, as in Figure $\mathrm{P} 4.20$. If the speed of the stream is $40.0 \mathrm{~m} / \mathrm{s}$, at what height will the water strike the building?

Key Concepts

Projectile Motion

Projectile motion refers to the two-dimensional motion of an object under the influence of gravity, where the only acceleration acting on the object is due to gravity. This concept explains how objects travel along a curved path, or trajectory, when launched with an initial velocity at an angle, and it requires analyzing horizontal and vertical components separately since they behave differently under gravity.

Decomposition of Velocity

Decomposing velocity involves breaking down the initial velocity into its horizontal and vertical components using trigonometric functions such as sine and cosine. This is crucial because the horizontal component remains constant (ignoring air resistance) while the vertical component is affected by gravitational acceleration, thus affecting the object’s trajectory.

Kinematic Equations

Kinematic equations describe how the position and velocity of an object change over time under constant acceleration. In projectile motion, these equations are used to determine the vertical displacement due to gravity and the time it takes for the projectile to cover a horizontal distance, enabling the calculation of where and when the projectile will reach a specific point.

Time of Flight Determination

Determining the time of flight is achieved by using the horizontal component of velocity and the horizontal distance to the target. Since the horizontal motion has constant velocity, the time can be calculated by dividing the horizontal distance by the horizontal velocity, which is critical for subsequently determining the vertical position at that time.

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