Free Fall Problem: Time, Distance, And Initial Speed

Dec 1, 2025 by Tom Lembong 53 views

Let's break down this classic physics problem step by step. We're dealing with a body in free fall, which means the only force acting on it is gravity (we're neglecting air resistance here, guys!). We know the velocities at two points, A and B, and we need to find the time it takes to travel between them, the distance between them, and the initial velocity of the object.

Understanding Free Fall

Before we dive into the calculations, let's quickly recap the key concepts of free fall. The acceleration due to gravity, denoted as 'g', is approximately 9.8 m/s². This means that the velocity of the object increases by 9.8 meters per second every second it falls. The equations of motion for constant acceleration will be our best friends in solving this problem. These equations relate displacement, initial velocity, final velocity, acceleration, and time. Remember, in free fall, the acceleration is constant and equal to 'g'. Understanding these concepts is crucial for tackling not just this problem, but any problem related to kinematics. Free fall is a simplified model that helps us understand the effects of gravity on moving objects. This simplification allows us to use well-defined equations and focus on the core physics principles. In reality, air resistance can significantly affect the motion of falling objects, especially at higher speeds. However, for many introductory physics problems, neglecting air resistance provides a good approximation and simplifies the calculations considerably. It's important to keep in mind the limitations of this model and understand when it's appropriate to use it.

Part A: Time to Travel Between A and B

First, let's find the time it takes for the body to travel from point A to point B. We know the velocities at both points: v_A = 22 m/s and v_B = 35 m/s. We also know the acceleration due to gravity, g = 9.8 m/s². We can use the following equation of motion:

v_B = v_A + gt

Where:

v_B is the final velocity (35 m/s)
v_A is the initial velocity (22 m/s)
g is the acceleration due to gravity (9.8 m/s²)
t is the time we want to find

Let's rearrange the equation to solve for t:

t = (v_B - v_A) / g

Now, plug in the values:

t = (35 m/s - 22 m/s) / 9.8 m/s²

t = 13 m/s / 9.8 m/s²

t ≈ 1.33 seconds

Therefore, it takes approximately 1.33 seconds for the body to travel from point A to point B. Make sure you use the correct units throughout the calculation. A common mistake is mixing up units, which can lead to incorrect results. In this case, we are using meters for distance and seconds for time, which are consistent with the standard unit system (SI units). Always double-check your units to avoid errors and ensure that your answer makes sense in the context of the problem.

Part B: Distance Between A and B

Now, let's calculate the distance between points A and B. We can use another equation of motion:

d = v_A * t + (1/2) * g * t²

Where:

d is the distance we want to find
v_A is the initial velocity (22 m/s)
t is the time (1.33 s)
g is the acceleration due to gravity (9.8 m/s²)

Plug in the values:

d = (22 m/s) * (1.33 s) + (1/2) * (9.8 m/s²) * (1.33 s)²

d = 29.26 m + (4.9 m/s²) * (1.77 s²)

d = 29.26 m + 8.67 m

d ≈ 37.93 meters

So, the distance between points A and B is approximately 37.93 meters. Notice how we used the time we calculated in the previous step. This is a common strategy in solving physics problems: break the problem down into smaller parts and use the results from previous calculations to solve subsequent parts. This approach can make complex problems more manageable and reduce the chance of errors. Always pay attention to the relationships between different variables and how they affect each other. Understanding these relationships is key to developing a strong intuition for physics.

Alternatively, we could have used the following equation:

v_B² = v_A² + 2gd

Rearranging for d:

d = (v_B² - v_A²) / (2g)

d = (35² - 22²) / (2 * 9.8)

d = (1225 - 484) / 19.6

d = 741 / 19.6

d ≈ 37.81 meters

This slight difference is due to rounding errors in the previous calculation. Both methods are valid, and the small difference highlights the importance of considering rounding errors in numerical calculations.

Part C: Initial Speed

To find the initial speed, we need more information. The problem only provides the speeds at points A and B after the object has already been in free fall. We can determine the initial velocity if we know the time or distance from the point where the object was released to point A. Let's assume we want to find the velocity at time t=0 (the initial velocity). We can denote the velocity at point A as v_A and the time it took to reach point A from the start as t_A. Then we can use the following equation:

v_A = v_0 + gt_A

Where:

v_A is the velocity at point A (22 m/s)
v_0 is the initial velocity (what we want to find)
g is the acceleration due to gravity (9.8 m/s²)
t_A is the time it took to reach point A from the start

Rearranging for v_0:

v_0 = v_A - gt_A

Without knowing t_A, we cannot find v_0. Let's say, hypothetically, that it took 1 second to reach point A from the start (t_A = 1 s). Then:

v_0 = 22 m/s - (9.8 m/s²) * (1 s)

v_0 = 22 m/s - 9.8 m/s

v_0 = 12.2 m/s

So, if it took 1 second to reach point A, the initial velocity would be 12.2 m/s. If we are given another piece of information, such as the distance traveled before reaching point A, we could use a similar approach with a different kinematic equation. Always identify the knowns and unknowns in the problem and choose the appropriate equation to solve for the desired variable.

Summary

We successfully calculated the time it took to travel between points A and B, the distance between them, and discussed how to calculate the initial velocity given additional information. Remember to use the correct equations of motion, pay attention to units, and break down complex problems into smaller, more manageable steps. Keep practicing, and you'll become a free-fall master in no time! Remember, physics is all about understanding the relationships between different concepts, so don't just memorize formulas – try to understand why they work and how they apply to real-world situations. Good luck, guys!