Question

An object whose mass is \(25 \mathrm{~kg}\) is projected upward from the surface of the earth with an initial velocity of \(60 \mathrm{~m} / \mathrm{s}\). The only force acting on the object is the force of gravity. Plot the velocity of the object versus elevation. Determine the elevation of the object, in \(\mathrm{ft}\), when its velocity reaches zero. The acceleration of gravity is \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\).

Short answer

The elevation at which the velocity reaches zero is approximately 602.63 ft.

Step-by-step solution

Identify known values

We know the mass of the object is 25 kg, initial velocity is 60 m/s, and acceleration due to gravity is 9.8 m/s². We need to find the elevation at which the velocity becomes zero.

Use the kinematic equation

The kinematic equation that relates velocity, initial velocity, acceleration, and displacement is: \[v^2 = u^2 + 2a s\]. Set \(v = 0\) (velocity reaches zero), \(u = 60 \mathrm{\;m/s}\), and \(a = -9.8 \, \mathrm{m/s^2}\) (since gravity acts in the opposite direction).

Solve for displacement (elevation)

Plug in the values into the kinematic equation: \[0 = (60)^2 + 2(-9.8) s\] Simplifying this: \[0 = 3600 - 19.6s\] \[19.6s = 3600\] \[s = \frac{3600}{19.6}\] el{\wich we

Convert displacement to feet

Since 1 meter is equivalent to approximately 3.28084 feet, multiply the elevation by this conversion factor: \[183.67 \, \text{m} \, \times 3.28084 \, \text{ft/m} \approx 602.63 \, \text{ft}\]

Key concepts

Projectile Motion

Projectile motion refers to the motion of an object that is thrown or projected into the air and is subject only to the force of gravity. Understanding this concept helps us determine the object's path or trajectory. When analyzing projectile motion, it is crucial to separate the object's vertical and horizontal components.

Key points to consider:

• Initial Velocity: The speed at which the object is thrown or projected, in our example, it's 60 m/s.
• Acceleration: Due to gravity, which is approximately 9.8 m/s² near the earth's surface.
• Displacement: The change in the position of the object, which we calculate to find out how high the object travels.

By separating the vertical from the horizontal components of motion, we can use kinematic equations to solve for various parameters such as the height the object reaches and the time it takes to return to the ground.

Gravity

Gravity is the force that attracts objects toward the center of the Earth. It plays a significant role in projectile motion. Here are a few important aspects of gravity:

• Gravity acts downwards, which means it will always oppose upward motion and accelerate downward motion.
• The acceleration due to gravity on Earth is approximately 9.8 m/s². This value is used in kinematic equations to calculate various parameters of motion.
• In our exercise, gravity decelerates the object as it moves upwards, reducing its velocity until it reaches zero at the highest point.

By understanding the influence of gravity, one can predict the motion of an object in free fall. This includes both the ascent and descent phases of the object's trajectory.

Velocity vs. Time Graph

A velocity vs. time graph is a helpful tool to visualize the motion of an object. For projectile motion, especially vertical motion, the graph provides insights into how the object's speed changes over time.

Important points to understand:

• The slope of the graph represents acceleration. In our case, the acceleration due to gravity is constant at -9.8 m/s², so the slope of the graph is a straight line going downwards.
• The y-intercept represents the initial velocity. For our example, the initial velocity is 60 m/s.
• When the velocity reaches zero, it indicates the highest point of the object's trajectory. Beyond this point, the velocity will become negative, showing that the object is falling back to the ground.

By plotting a velocity vs. time graph, one can easily interpret how quickly an object decelerates as it rises and accelerates as it falls due to gravity. This reinforces the understanding of concepts like initial velocity, maximum height, and the period of ascent and descent.