Question

A projectile of mass \(m\) is launched vertically upward from ground level at time \(t=0\) with initial velocity \(v_{0}\) and is acted upon by gravity and air resistance. Assume the drag force is proportional to velocity, with drag coefficient \(k\). Derive an expression for the time, \(t_{m}\), when the projectile achieves its maximum height.

Short answer

Answer: The expression for the time when the projectile reaches its maximum height is given by: $$t_m = -\frac{m}{k} \ln \left(\frac{mg}{kv_0}\right)$$ where \(m\) is the mass of the projectile, \(k\) is the drag coefficient, \(g\) is the gravitational acceleration, and \(v_0\) is the initial velocity.

Step-by-step solution

Identify the relevant equations and variables

Let's first identify the relevant equations and variables that we'll use to derive the expression for the maximum height time. We're given that the mass of the projectile is \(m\), the initial velocity is \(v_0\), the drag coefficient is \(k\), and the gravitational acceleration is \(g\). The equation we'll use is Newton's second law: $$F = ma$$ In this case, the forces acting on the projectile are gravity and air resistance (drag force), which can be expressed as: $$F_{g} = -mg$$ $$F_{d} = -kv$$

Write the net force equation

Now that we've identified the forces acting on the projectile, let's write the net force equation. Since the projectile is launched vertically upward, we'll write it for the vertical direction: $$ma = F_{g} + F_{d}$$ Substitute the expressions for the gravitational and drag forces: $$m \frac{dv}{dt} = -mg - kv$$

Separate variables and integrate

Now we'll separate the variables and integrate to find the velocity of the projectile as a function of time. $$\frac{m}{mg + kv} dv = dt$$ Integrating both sides: $$\int_{v_0}^v \frac{m}{mg + kv} dv = \int_0^t dt$$

Determine the velocity expression

Solve the integrals and determine the expression for velocity as a function of time: $$-m \ln \left( \frac{mg + kv}{mg + kv_0} \right) = t$$ Solve for \(v\): $$v(t) = \frac{mg + kv_0 e^{-\frac{k}{m}t}}{k} - \frac{mg}{k}$$

Find the time when the projectile reaches its maximum height

To find the time \(t_m\) when the projectile achieves its maximum height, we need to find when the velocity \(v(t)\) is equal to zero. Set \(v(t) = 0\) and solve for t: $$\frac{mg + kv_0 e^{-\frac{k}{m}t}}{k} - \frac{mg}{k} = 0$$ Multiply both sides by \(k\) and add \(\frac{mg}{k}\): $$kv_0 e^{-\frac{k}{m}t} = mg$$ Divide both sides by \(kv_0\) and take the natural logarithm: $$-\frac{k}{m}t_m = \ln \left(\frac{mg}{kv_0}\right)$$ Finally, solve for \(t_m\) to get the expression for the time when the projectile reaches its maximum height: $$t_m = -\frac{m}{k} \ln \left(\frac{mg}{kv_0}\right)$$

Key concepts

Projectile Motion with Air Resistance

Projectile motion in physics usually deals with the motion of an object that is thrown or propelled through the air and is subject to gravitational forces. However, when air resistance is considered, the scenario becomes more complex. Air resistance, also known as drag, is the force exerted by air against the movement of the projectile. It is often proportional to the velocity of the object, making the resulting motion difficult to analyze and predict.

The effect of air resistance on a projectile can be dramatic. With air resistance, the path of the projectile is no longer a simple parabola, and the maximum height and range are both reduced. To accurately predict the motion, we must include this resistive force in our calculations, as seen in the step by step solution where the drag coefficient, denoted by 'k', directly affects the equation of motion.

Understanding projectile motion with air resistance is essential for many real-world applications, such as ballistics, sports, and engineering. In educational contexts, it provides a fertile ground for applying differential equations and integrating concepts from physics to solve practical problems.

Newton's Second Law Application

Newton's Second Law of Motion states that the force exerted on an object is equal to the mass of the object multiplied by its acceleration:
\[ F = ma \].

When applying this law to projectile motion with air resistance, we consider two main forces: the gravitational force pulling the projectile downward, and the drag force working against the projectile's motion. In the given exercise, the forces due to gravity and air resistance are combined to form the net force acting on the projectile. Newton's Second Law is the starting point for creating an equation that represents this net force and is then used to derive more detailed information about the motion, such as velocity and time to reach maximum height, through further analysis and integration.

This application of Newton's Second Law provides a link between basic physical principles and their mathematical description, showcasing the interplay between theory and practice in physics.

Separation of Variables

Separation of variables is a common mathematical method used to solve ordinary differential equations. This technique involves rearranging the equation so that each variable and its differential are on opposite sides of the equation. The goal is to obtain an equation where one side depends only on one variable and the other side depends only on the other variable, allowing us to integrate both sides independently.

In the context of our projectile motion problem, separation of variables was used to dissociate the variables of velocity and time in the differential equation \( m\frac{dv}{dt} = -mg - kv \).

Rewriting it into the separable form and integrating both sides allows us to find the velocity of the projectile as a function of time. This method provides a straightforward and methodical approach to tackle differential equations that appear in various scientific fields, from physics and engineering to ecology and economics.

Integrating Velocity to Time

Once the variables in the differential equations are separated, the next step is to integrate these expressions to get a relationship between velocity and time. Through integration, we can get a function that describes how velocity changes over time. The act of integrating is essentially summing up tiny changes over an interval, which, in our problem, is the change in velocity over time due to the forces acting on the projectile.

In the case of our projectile affected by air resistance, integrating the separated variables from the initial velocity to any given velocity gives us an integral on the left side of the equation. On the right side, we have the integral of dt from 0 to t, which expresses the accumulation of time as the projectile ascends. The result of this integration process is an equation for velocity as a function of time, which can then be analyzed further to find out when the velocity is zero, indicating the time at which the projectile reaches its maximum height.

This integration process is not only fundamental in solving problems in physics but also serves as a valuable tool in many areas of applied mathematics.