Problem 177 - Calculus Volume 3

Question

A projectile is shot in the air from ground level with an initial velocity of \(500 \mathrm{~m} / \mathrm{sec}\) at an angle of \(60^{\circ}\) with the horizontal. The graph is shown here: What is the total flight time of the projectile?

Step-by-Step Solution

Answer

The total flight time of the projectile is approximately 88.41 seconds.

Step 1: Resolve Initial Velocity into Components

First, resolve the initial velocity of the projectile into horizontal and vertical components. Use the angle given to find these components. The initial velocity is given as \( 500 \text{ m/s} \) and the angle is \( 60^\circ \).The horizontal component \( v_{x} \) is calculated using cosine:\[v_{x} = 500 \cdot \cos(60^\circ) = 250 \text{ m/s}.\]The vertical component \( v_{y} \) is calculated using sine:\[v_{y} = 500 \cdot \sin(60^\circ) = 250\sqrt{3} \approx 433 \text{ m/s}.\]

Step 2: Calculate Time of Flight Using Vertical Motion

Calculate the time of flight using the vertical motion of the projectile. The vertical motion is influenced by gravity, which decelerates the upward motion initially and accelerates the downward motion.The formula for time of flight \( T \) considering the projectile returns to the same vertical position (ground level) is:\[ T = \frac{2v_{y}}{g}, \]where \( g = 9.8 \text{ m/s}^2 \) is the acceleration due to gravity.Substitute the vertical component calculated:\[ T = \frac{2 \times 250\sqrt{3}}{9.8} = \frac{500\sqrt{3}}{9.8} \approx 88.41 \text{ seconds}. \]

Key Concepts

Initial Velocity

When a projectile is launched, its ""initial velocity"" refers to the speed and direction at which it starts moving. In this exercise, the initial velocity of the projectile is given as 500 meters per second. This represents how fast and in which direction an object begins its motion right from the ground level.

• The initial velocity is crucial because it sets the stage for how the projectile will move through its entire flight.
• In physics problems, this is often accompanied by an angle of projection.
• For this exercise, the angle is 60 degrees from the horizontal.

Together, these two components define the starting point of a projectile's journey.

Horizontal and Vertical Components

Initial velocity is often broken down into "horizontal" and "vertical components". This separation helps in analyzing and understanding projectile motion more easily:

• **Horizontal Component:** This represents the speed of the projectile along the horizontal axis. It's calculated using the cosine of the angle. For our problem, it results in a velocity of 250 m/s.


• **Vertical Component:** This describes how fast the projectile is moving vertically. Use the sine of the angle to calculate it. In our case, it results in approximately 433 m/s.

These components help predict how the projectile will behave under the effects of gravity and time.

Time of Flight

The "time of flight" refers to the total time the projectile remains in the air from launch until it hits the ground again. It depends significantly on the vertical component of initial velocity and gravity.

• To find this, we rely on vertical motion, since gravity acts vertically.
• The formula used is \( T = \frac{2v_{y}}{g} \), where \(v_y\) is the vertical component, and \(g\) is the acceleration due to gravity.
• Plugging in values for our example, \( T = \frac{500\sqrt{3}}{9.8} \approx 88.41\) seconds.

This tells us how long the projectile takes to land back to its initial vertical position, which is crucial for planning and analysis.

Acceleration due to Gravity

"Acceleration due to gravity" is a constant that affects all objects moving under the influence of Earth's gravitational pull.

• It is denoted by \(g\) and equals approximately 9.8 meters per second squared \(\text{m/s}^2\).
• In projectile motion, it acts only on the vertical component, slowing down upward motion and speeding up the downward motion.


• This value is integral to determining the time of flight and overall trajectory of the projectile.

Regardless of other variables, gravity ensures the projectile will eventually return to the ground.