Question

A skier launches off a ski jump with a horizontal velocity of \(30.0 \mathrm{~m} / \mathrm{s}\) (and no vertical velocity component). What are the magnitudes of the horizontal and vertical components of her velocity the instant before she lands 2.00 s later?

Short answer

Answer: The magnitude of the horizontal component of velocity is 30.0 m/s, and the magnitude of the vertical component of velocity is 19.62 m/s (downward).

Step-by-step solution

Find the horizontal component of velocity just before landing

Since the horizontal component of velocity remains constant in the absence of horizontal acceleration, the magnitude of the horizontal component of her velocity just before landing is the same as her initial horizontal velocity: Horizontal velocity component \(V_{x_f} = V_{x_i} = 30.0 \, \mathrm{m/s}\)

Calculate the final vertical velocity component

To find the vertical component of velocity just before the skier lands, let's use the constant acceleration equation: \(V_{y_f} = V_{y_i} + gt\) where \(V_{y_f}\) is the final vertical velocity component, \(V_{y_i}\) is the initial vertical velocity component, \(g\) is the acceleration due to gravity (negative for downward direction), and \(t\) is the time elapsed. Here, \(V_{y_i} = 0\), \(g = -9.81 \, \mathrm{m/s²}\), and \(t = 2.00 \, \mathrm{s}\). Plug these values into the equation: \(V_{y_f} = 0 + (-9.81 \, \mathrm{m/s²}) (2.00 \, \mathrm{s})\) \(V_{y_f} = -19.62 \, \mathrm{m/s}\) So, the magnitude of the vertical component of her velocity just before she lands is: Vertical velocity component \(|V_{y_f}| = 19.62 \, \mathrm{m/s}\) (downward) #Summary# At the instant before the skier lands 2.00 seconds later, - The magnitude of the horizontal component of velocity is 30.0 m/s. - The magnitude of the vertical component of velocity is 19.62 m/s (downward).

Key concepts

Horizontal Velocity

When we talk about horizontal velocity in projectile motion, we're focusing on the movement parallel to the ground. It's crucial to understand that, unless acted upon by an external horizontal force (like air resistance, which is often negligible in many physics problems), the horizontal component of velocity remains unchanged throughout the projectile's flight. This is because once the object is in the air, no horizontal acceleration is acting on it to change its speed.

For example, in our skier scenario, the initial horizontal velocity is given as 30.0 m/s, and because there's no horizontal acceleration, this velocity doesn't change. So just before landing, the skier still moves horizontally at 30.0 m/s. What's important to note is that the constant horizontal velocity is independent of any vertical motion the skier undergoes.

Vertical Velocity

Vertical velocity, on the other hand, is concerned with the component of motion that is perpendicular to the ground. The key factor influencing vertical velocity in projectile motion is gravity, which provides a constant downward acceleration. Unlike horizontal motion, vertical motion does change over time due to this acceleration.

In our skier's case, she starts with a zero initial vertical velocity, as she launches off horizontally. Over time, gravity pulls her downwards, increasing her vertical velocity at a predictable rate until she reaches the ground. The final vertical velocity can be determined using the kinematic equations for constant acceleration, which in this particular situation result in a downward vertical velocity of 19.62 m/s just before landing.

Constant Acceleration

Constant acceleration is a fundamental concept in kinematics, referring to a steady change in velocity. In projectile motion, the only source of constant acceleration is typically gravity, acting at approximately 9.81 m/s² downwards near the Earth's surface. This acceleration is the reason why objects pick up speed as they fall.

Because gravity is a constant force and acts only in the vertical direction, it doesn't affect horizontal velocity. Thus, constant acceleration in projectile motion is always associated with vertical velocity. Despite starting without vertical movement, our skier eventually develops a significant vertical velocity due to the unyielding pull of gravity during her 2-second flight.

Kinematics Equations

Kinematics equations provide the mathematical tools to describe motion with constant acceleration, without needing to consider the forces that cause such motion. There are four main kinematic equations, each useful for different scenarios based on known and unknown variables.

In projectile motion, these equations help us calculate various aspects like displacement, velocity, and time. Specifically for finding the skier's final vertical velocity, we used the equation \(V_{y_f} = V_{y_i} + gt\), where \(V_{y_f}\) is the final vertical velocity, \(V_{y_i}\) is the initial vertical velocity (zero in our case), \(g\) is the acceleration due to gravity (negative as it's directed downwards), and \(t\) is the time elapsed. By understanding and applying these equations correctly, we can predict the skier's vertical velocity at any point after launch, including just before she lands.