Vertical Motion and Free Fall: Videos & Practice Problems

Vertical motion problems are solved using the same principles as horizontal motion, focusing on free fall where gravity is the only acting force. Objects in free fall experience a constant acceleration of g=9.8m/s². When analyzing these problems, it's essential to identify five variables, using Δy for displacement and v for velocity. The acceleration a is typically negative when the upward direction is considered positive, simplifying calculations.

Vertical Motion & Free Fall

Understanding vertical motion is crucial for solving problems involving objects moving up or down. Vertical motion problems utilize the same principles and equations as horizontal motion, but with some key differences. A fundamental concept in vertical motion is free fall, which occurs when the only force acting on an object is gravity, denoted as F_g. For instance, if you hold a box with a string, both tension and gravity act on it, meaning it is not in free fall. However, if you release the box, it enters free fall as gravity is the sole force acting on it.

Even when an object is thrown upwards, it is still in free fall once it is released, as gravity remains the only acting force. This is significant because objects in free fall experience a constant vertical acceleration, which allows us to apply the same equations of uniformly accelerated motion (UAM) to vertical motion problems. The key difference lies in the variables used: while horizontal motion uses Δx and v_x, vertical motion uses Δy and v_y.

In vertical motion, the acceleration due to gravity, represented as g, is a constant value of 9.8 m/s² on Earth. This means that regardless of the object's weight, all objects in free fall accelerate downwards at this rate. The acceleration term in equations can be either positive or negative, depending on the chosen direction of motion. A common approach is to define upward as positive, which results in a negative acceleration of -9.8 m/s² for free-falling objects.

To solve a vertical motion problem, one typically starts by drawing a diagram and identifying the five key variables: initial velocity (v₀), final velocity (v_y), displacement (Δy), acceleration (a_y), and time (Δt). For example, if a ball is dropped from a height of 100 meters, the initial velocity is 0, and the displacement is -100 meters (since it falls downwards). The acceleration is -9.8 m/s².

Using the second equation of motion, v_y² = v₀² + 2a_yΔy, we can find the final velocity just before the ball hits the ground. Plugging in the values, we have:

v_y² = 0 + 2(-9.8)(-100)

This simplifies to v_y² = 1960, leading to v_y = ±√1960, which gives approximately ±44.3 m/s. Since the ball is falling downwards, the final velocity is -44.3 m/s, indicating the direction of motion.

In summary, vertical motion problems can be approached similarly to horizontal motion, with careful attention to the direction of forces and the application of the correct equations. Understanding the role of gravity and the concept of free fall is essential for accurately solving these problems.

A rock is thrown vertically upward with a speed of 27.0 m/s from the roof of a 31.0-m-tall building. The rock doesn't hit the building on its way back down and lands in the street below. (a) What is the speed of the rock just before it hits the street? (b) How much time elapses from when the rock is thrown until it hits the street?

• A. (a) vy = 36.6 m/s downward;
  (b) t = 3.88 s
• B. (a) vy = 11.0 m/s downward;
  (b) t = 3.88 s
• C. (a) vy = 37.0 m/s downward;
  (b) t = 34.2 s
• D. (a) vy = 36.6 m/s downward;
  (b) t = 6.49 s

Answer: (D)

A student throws a set of keys vertically upward to her sorority sister who is in a window 14.00 m above. The second student catches the keys 1.50 s later. (a) With what initial velocity were the keys thrown? (b) What was the velocity of the keys just before they were caught?

• A. (a) v_0y=37.5 m/s
  (b) v_y=22.8 m/s
• B. (a) v_0y=16.7 m/s
  (b) v_y=2 m/s
• C. (a) v_0y=2 m/s
  (b) v_y=0 m/s

Answer: (B)

Here’s what students ask on this topic:

What is the acceleration due to gravity in free fall?

The acceleration due to gravity in free fall is denoted by the symbol g and has a constant value of 9.8 m/s² on Earth. This means that any object in free fall, regardless of its mass, will accelerate downwards at this rate, assuming no other forces (like air resistance) are acting on it. This constant acceleration is crucial for solving vertical motion problems using the kinematic equations.

How do you determine the direction of acceleration in vertical motion problems?

In vertical motion problems, the direction of acceleration is determined by the choice of the positive direction. If you choose the upward direction as positive, then the acceleration due to gravity a_y will be negative, i.e., -9.8 m/s². Conversely, if you choose the downward direction as positive, then a_y will be positive, i.e., 9.8 m/s². Consistently applying this choice simplifies calculations and helps avoid confusion.

What are the key variables used in solving vertical motion problems?

The key variables used in solving vertical motion problems are: initial velocity (v₀), final velocity (v), displacement (Δy), acceleration (a_y), and time (t). These variables are used in the kinematic equations to solve for unknowns. For vertical motion, displacement is denoted as Δy and acceleration is typically ±9.8 m/s² depending on the chosen positive direction.

How do you solve for the final velocity of an object in free fall?

To solve for the final velocity (v) of an object in free fall, you can use the kinematic equation: $v^2=v^0+2ay\mathrm{\Delta y}$. Here, v₀ is the initial velocity, a_y is the acceleration due to gravity (typically -9.8 m/s² if upward is positive), and Δy is the displacement. Plug in the known values and solve for v.

What is free fall and how does it differ from other types of motion?

Free fall is a type of motion where the only force acting on an object is gravity. This means that the object experiences a constant acceleration of 9.8 m/s² downwards. Unlike other types of motion where multiple forces might be acting (e.g., tension, friction), free fall is characterized by the absence of any force other than gravity. This simplifies the analysis and allows the use of kinematic equations with a constant acceleration.