Question

What is the maximum height above ground a projectile of mass $0.79\mathrm{kg}$, launched from ground level, can achieve if you are able to give it an initial speed of $80.3\mathrm{m}/\mathrm{s}?$

Short answer

Answer: The maximum height the projectile can achieve is approximately 261.52 meters above ground level.

Step-by-step solution

Write down the conservation of energy equation

Since the energy is conserved, the initial kinetic energy (KE) will equal the potential energy (PE) at the maximum height. We can write this as: $K{E}_{initial}=P{E}_{max}$

Write the expressions for initial kinetic energy and potential energy at the maximum height

The initial kinetic energy can be expressed as: $K{E}_{initial}=\frac{1}{2}m{v}^2$ And potential energy at the maximum height can be expressed as: $P{E}_{max}=mg{h}_{max}$ Here, $m$ is the mass of the projectile, $v$ is the initial speed, $g$ is the acceleration due to gravity, and $h_{max}$ is the maximum height.

Substitute the expressions and given values into the conservation of energy equation

Recall the conservation of energy equation we wrote in Step 1: $K{E}_{initial}=P{E}_{max}$ Now, substitute the expressions from Step 2 and the given values of mass and initial speed: $\frac{1}{2}(0.79\mathrm{kg})(80.3\mathrm{m}/\mathrm{s}{)}^2=(0.79\mathrm{kg})(9.81\mathrm{m}/{\mathrm{s}}^2)h_{max}$

Solve for the maximum height $h_{max}$

Now we only have one unknown in the equation, which is $h_{max}$. Solve for $h_{max}$: $h_{max}=\frac{\frac{1}{2}(0.79)(80.3{)}^2}{(0.79)(9.81)}$ $h_{max}\approx 261.52\mathrm{m}$ Thus, the maximum height the projectile can achieve is approximately 261.52 meters above ground level.

Key concepts

Conservation of Energy

The concept of Conservation of Energy is fundamental in understanding projectile motion and many other physics phenomena. Energy cannot be created or destroyed; it can only change forms. This principle allows us to track energy as it transforms from one type to another. In the context of projectile motion:

• We begin with kinetic energy when the projectile is launched.
• This energy transforms into potential energy as the projectile gains height.

At the highest point of its trajectory, the projectile has maximum potential energy and minimum kinetic energy. This energy transformation is crucial to predicting the projectile's behavior, such as its maximum height. The conservation equation $$K{E}_{initial}=P{E}_{max}$$uses this principle, illustrating how initial kinetic energy directly relates to the height achieved.

Kinetic Energy

Kinetic Energy is the energy associated with motion. When a projectile is launched, it possesses kinetic energy due to its speed. This is calculated using the formula:$$KE=\frac{1}{2}m{v}^2$$where:

• $m$ is the mass of the projectile.
• $v$ is its velocity.

The initial kinetic energy is crucial for determining how much energy is available to be converted into potential energy. For example, in our problem, the projectile's initial speed of 80.3 m/s gives it significant kinetic energy, allowing it to reach a high point in its trajectory. As the projectile rises, this kinetic energy decreases while potential energy increases, demonstrating the conservation of energy principle.

Potential Energy

Potential Energy represents the energy stored due to an object's position or configuration. For a projectile reaching a height, the potential energy is given by:$$PE=mgh$$where:

• $m$ is the mass of the object.
• $g$ is the acceleration due to gravity, approximately 9.81 m/s${}^2$.
• $h$ is the height above ground.

In our exercise, as the projectile ascends, its potential energy increases as it gains height. At its maximum height, all the initial kinetic energy has been transformed into potential energy. This conversion is complete, leaving the projectile momentarily at rest before gravity pulls it back down. Calculating the maximum height involves ensuring that every bit of initial kinetic energy is accounted for as potential energy.

Physics Problem Solving

Physics Problem Solving skills are essential when tackling exercises involving concepts like projectile motion. To solve such problems:

• Start by identifying what you know and what you need to find out.
• Use relevant formulas that tie together the known quantities and the unknowns.
• Substitute values and solve algebraically, keeping track of units for consistency.

In this projectile problem, we used the conservation of energy equation effectively. By substituting the known mass and speed, we solved for the unknown maximum height. Breaking down problems into smaller, manageable steps helps enhance comprehension and ensures accurate solutions. Remember, practicing these steps hones your skills to tackle even more complex problems efficiently.