91Ó°ÊÓ

In projectile motion, understanding how to decompose an initial velocity into horizontal and vertical components is crucial. It involves using the original velocity of an object at a given angle and breaking it down into two parts.
The horizontal component is determined by multiplying the initial velocity by the cosine of the given angle. For example, if an object is thrown with an initial velocity of 11 m/s at an angle of 65 degrees, the horizontal component would be \(v_{0x} = 11 \cos 65^{\circ} \approx 4.64 \text{ m/s}\).
Similarly, the vertical component can be found by using the sine of the angle. In the same example, this would be \(v_{0y} = 11 \sin 65^{\circ} \approx 9.97 \text{ m/s}\).
These components are useful in solving various parts of projectile motion problems, allowing you to predict how long the projectile will be in the air and how far it will travel.

The time of flight is the duration a projectile is in the air. It is crucial to know this when analyzing projectile motion, as it affects other parameters like range and maximum height.
For the kit in our example, the time of flight is when the climber catches it, which is when the vertical velocity is zero. To find this, use the formula derived from kinematic equations: \(v_y = v_{0y} - gt\), where \(v_y\) is the final vertical velocity, \(g\) is the acceleration due to gravity (approximately \(9.8 \, \text{m/s}^2\)), and \(t\) is the time.
By setting the final vertical velocity \(v_y\) to 0 (since the kit is caught at zero vertical speed), the time can be calculated. Solving \(0 = 9.97 - 9.8t\) gives \(t \approx 1.017 \text{ seconds}\).
This time of flight is then used to determine other quantities such as the vertical displacement.

Kinematic equations play a vital role in describing motion. They help calculate displacement, velocity, and time under constant acceleration. In projectile motion, these equations allow us to predict the behavior of objects thrown through the air.
One key equation used is for vertical motion: \(h = v_{0y}t - \frac{1}{2}gt^2\). This formula helps calculate the vertical displacement or the change in height during flight. In this equation:

• \(h\) represents vertical height or displacement.
• \(v_{0y}\) is the initial vertical speed.
• \(g\) is the acceleration due to gravity.
• \(t\) is the time of flight.

Substituting the known values into the equation helps solve complex problems in a step-by-step manner, like determining how high the first aid kit needed to travel between the two climbers.

Vertical displacement refers to how far an object has moved vertically during its flight. In our scenario, this represents the difference in height between the two climbers.
To find this, use the kinematic equation for vertical displacement: \(h = v_{0y}t - \frac{1}{2}gt^2\). This formula balances the initial upward motion and the downward pull of gravity over time.
With \(v_{0y} = 9.97 \text{ m/s}\) and \(t \approx 1.017 \text{ seconds}\), substituting these values gives \(h \approx 9.97 \times 1.017 - \frac{1}{2} \times 9.8 \times (1.017)^2\).
This calculation yields a vertical height of approximately 5.09 meters. Therefore, the first aid kit needs to travel this vertical distance between the two climbers.