Question

A ball starts from rest and rolls down a hill with uniform acceleration, traveling 200 m during the second 5.0 s of its motion. How far did it roll during the first 5.0 s of motion?

Short answer

The ball rolled 66.67 meters during the first 5 seconds.

Step-by-step solution

Understanding the Problem

A ball rolls down a hill with uniform acceleration. It covers 200 meters during the second 5 seconds of its motion. We need to find out how far it rolls during the first 5 seconds of its motion.

Identify Known Variables

The known quantities are the distance traveled during the second 5 seconds, which is 200 m, and the time period of interest being the first 5 seconds.

Apply the Uniform Acceleration Formula

For an object starting from rest (initial velocity, \( u = 0 \)), the distance \( s \) covered in time \( t \) under uniform acceleration \( a \) is given by the equation: \( s = ut + \frac{1}{2}at^2 \). Simplifying since \( u = 0 \), we have \( s = \frac{1}{2}at^2 \).

Find Distance Covered in First 10 Seconds

The object moves a distance \( s_1 \) in 5 seconds and \( s_2 \) between 5 and 10 seconds. The difference \( s_2 - s_1 = 200 \) m. The total distance after 10 seconds is \( \frac{1}{2}a(10)^2 \). Therefore: \( s_1 = \frac{1}{2}a(5)^2 \) and \( 200 = \frac{1}{2}a(10)^2 - \frac{1}{2}a(5)^2 \).

Solve for Acceleration

Solve the equation: \( 200 = \frac{1}{2}a(10^2 - 5^2) \). Simplifying gives \( 200 = \frac{1}{2}a(100 - 25) \) or \( 200 = \frac{1}{2}a(75) \). Thus, \( a = \frac{400}{75} \).

Calculate Distance in First 5 Seconds

Now using \( s_1 = \frac{1}{2}a(5)^2 \) and substituting the value of \( a \), we find \( s_1 = \frac{1}{2} \times \frac{400}{75} \times 25 \). This simplifies to \( s_1 = 66.67 \) m.

Key concepts

Uniform Acceleration

Uniform acceleration occurs when an object speeds up or slows down at a constant rate over time. This is an important concept in kinematics because it allows us to predict the future motion of objects when their acceleration doesn't change.
In a problem involving uniform acceleration like the exercise provided, we often use the formula:

• For an object starting from rest: \[ s = \frac{1}{2} a t^2 \] where:
  • \( s \) is the distance traveled,
  • \( a \) is the uniform acceleration,
  • \( t \) is the time elapsed.

Understanding and applying this formula is crucial for solving problems where the speed of the object is either increasing or decreasing at a constant rate. The key takeaway is to determine the acceleration before calculating how far an object travels in a given time.

Distance Traveled

In the context of physics and kinematics, the distance traveled by an object is the length of the path it takes during its motion. For objects moving with uniform acceleration, determining the distance involves knowing both the acceleration rate and the time period of interest.
In our given exercise, we are asked to find out the distance a ball rolls during the first 5 seconds under uniform acceleration. Compatibility with the formula for uniform acceleration means that once we compute the acceleration (using the data for the second 5-second interval), we can plug it back into our formula to find the distance for any other time frame.
Applying the formula \[ s_1 = \frac{1}{2} a t^2 \] for the first 5 seconds gives us the specific distance the ball traveled as it started from rest. Understanding how to isolate and plug-in known values helps solve these types of physics questions effectively.

Physics Problem Solving

Physics problem solving involves systematic steps, starting with understanding the problem and identifying known variables.
In our exercise, solving involved:

• Identifying variables like distance traveled during a defined time interval and knowing the object's start condition (rest in this case).
• Applying relevant formulas: Here, the formula for uniformly accelerated motion for an object starting from rest was applied.
• Algebraic manipulation: Equating known distances helped isolate unknown variables like acceleration.
• Iterative calculations: Using solved acceleration to then determine traveled distance in another interval.

The clarity in each step ensures correct setup of equations and helps avoid errors based on faulty initial assumptions or calculations.
This structured approach is crucial for success in both educational exercises and real-world physics applications.

Motion Dynamics

Motion dynamics delve into how an object's movement changes under the influence of forces. In kinematics, specifically when dealing with uniform acceleration, motion is typically analyzed in a linear path where forces like gravity are constant, such as our example with the rolling ball on a hill.

The constant influence of gravity accelerates the ball uniformly, and inherently affects how the calculations for distance and acceleration need to be conducted. Dynamics help explain the cause and effect relationship between applied forces and motion, emphasizing the need for understanding fundamentals like Newton's laws in more complex scenarios.
In simpler uniform motion scenarios, these dynamics help us derive formulas and principles like those used in our given problem set, making it possible to predict future movements of objects accurately.