Question

Show that the maximum height reached by a projectile is $$ y_{\max }=\left(v_{0} \sin \phi_{0}\right)^{2} / 2 g $$

Short answer

The maximum height formula \(y_{\max} = \left(v_{0} \sin \phi_{0}\right)^{2} / 2 g\) defines the maximum height reached by a projectile based on initial velocity, launch angle and gravitational force. This formula is derived from the second equation of motion and confirmed by substituting in the time at which the object reaches maximum height.

Step-by-step solution

Understanding given equation

The formula provided, \(y_{\max }=\left(v_{0} \sin \phi_{0}\right)^{2} / 2 g\), is the equation for the maximum height reached by a projectile. Here, \(v_0\) is the initial velocity, \(\phi_0\) is the initial angle at which the object is projected, \(g\) is the acceleration due to gravity, and \(y_{\max}\) is the maximum height reached by the projectile.

Applying the equation of motion

The second equation of motion states that R = \(v_{0} t - \frac{1}{2} g t^{2}\) when initial position is zero or where R is the displacement. At maximum height the final velocity (v) becomes zero. Using equation of motion \(v = v_0 - g t\) we find that the time at which object reaches at maximum height (\(t_{\max}\)) is \(t_{\max} = \frac{v_{0} \sin \phi_{0}}{g}\). Putting this into first equation we get \(y_{\max} = v_{0} (\frac{v_0 \sin \phi_0}{g}) - \frac{1}{2} g (\frac{v_0 \sin \phi_0}{g})^{2}\).

Simplifying the equation

When we simplify above equation we get \(y_{\max} = \left(v_{0} \sin \phi_{0}\right)^{2} / 2 g\), which matches the given equation.

Key concepts

Maximum Height

When we talk about projectile motion, maximum height refers to the peak point a projectile reaches in its path. This is crucial because it tells us how high the projectile has traveled before starting to descend.
In physics, we calculate this using the formula:
\[ y_{\text{max}} = \frac{(v_0 \sin \phi_0)^2}{2g} \]

• \(v_0\) - Initial velocity : The speed at which the projectile is launched.
• \(\phi_0\) - Launch angle : The angle at which the projectile is released relative to the horizontal.
• \(g\) - Acceleration due to gravity : Earth's gravitational pull, usually approximated as 9.81 m/s².

Understanding the formula means recognizing that the projectile's vertical component of velocity determines how high it will ascend before gravity overtakes its upward motion. The initial launch angle, especially when coupled with initial velocity, plays a key role in reaching the highest point.

Kinematic Equations

Kinematic equations are mathematical formulas used to describe the motion of a body under constant acceleration, like our projectile here. When dealing with projectiles, we often use the kinematic equation:
\[ R = v_0 t - \frac{1}{2} g t^2 \]
This formula helps in calculating displacement for a body moving under the influence of gravity, assuming the initial position is zero.
Here’s how the equation works:

• \(R\) is the displacement or change in position.
• The term \(v_0 t\) calculates the distance covered by the initial velocity over time \(t\).
• The subtraction \(-\frac{1}{2} g t^2\) accounts for the deceleration due to gravity.

This equation becomes particularly important when solving for the time it takes for a projectile to reach its maximum height and then fall back to a certain point.

Initial Velocity

Initial velocity \(v_0\) plays a pivotal role in determining the characteristics of projectile motion. This is the speed at the moment when the projectile is launched.
To find the maximum height of a projectile, the initial velocity must be broken into its components.

• Horizontal Component: \(v_0 \cos \phi_0\) — the constant velocity forward.
• Vertical Component: \(v_0 \sin \phi_0\) — the velocity that fights gravity upwards.

When the projectile reaches its maximum height, the vertical component of the velocity becomes zero before starting to descend. At this point, the initial velocity directly influences how high the projectile can rise, given the same initial launch angle.

Acceleration due to Gravity

Acceleration due to gravity is a fundamental force that acts on all objects with mass, pulling them towards the center of the Earth. In the context of projectile motion, it is pivotal.
This acceleration is symbolized by the letter \(g\) and is typically approximated as 9.81 m/s² on the surface of the Earth.

• Gravitational acceleration remains constant for most scenarios we encounter on Earth.
• This constant force decelerates a projectile's ascent and accelerates its descent.

Understanding gravity's role helps predict the time it takes to reach the maximum height and descend. In the absence of other forces, all projectiles experience this same acceleration, granting us a consistent parameter for calculating motion dynamics.