Calculating Initial Velocity A Sled Physics Problem

Hey everyone! Today, we're diving into a classic physics problem involving motion and acceleration. We'll break down a sled's velocity changes over time and figure out its initial speed. If you're just starting out in physics or need a refresher on kinematics, you've come to the right place. Let's get started!

Understanding the Problem

So, here's the deal. We have a sled moving along, and we're given some information about its velocity at different times. At 4 seconds, it's moving at a certain speed, which we'll call v₀. Then, at 9 seconds, its speed has increased to 15 m/s. We also know that the average acceleration during this time interval (from 4 seconds to 9 seconds) was 2 m/s². Our mission, should we choose to accept it (and we do!), is to calculate the value of that initial velocity, v₀. This is a typical kinematics problem, and we'll be using the concepts of average acceleration and uniform motion to solve it.

Key Concepts and Formulas

Before we jump into the calculations, let's quickly review the key concepts and the formula we'll be using. The most important concept here is average acceleration. Average acceleration is the rate at which the velocity of an object changes over a period of time. Mathematically, it's defined as the change in velocity divided by the change in time. The formula looks like this:

a_avg = (Δv) / (Δt)

Where:

• a_avg is the average acceleration
• Δv is the change in velocity (final velocity minus initial velocity)
• Δt is the change in time (final time minus initial time)

In our case, we know a_avg, the final velocity, the final time, and the initial time. We're trying to find the initial velocity. So, we'll need to rearrange this formula a little bit to solve for v₀.

Breaking Down the Problem

To make things crystal clear, let's identify the information we have and what we're looking for:

• Initial time (t₀): 4 seconds
• Final time (t): 9 seconds
• Velocity at t: 15 m/s
• Average acceleration (a_avg): 2 m/s²
• Initial velocity (v₀): This is what we want to find!

Now that we have all the pieces of the puzzle, we can start putting them together to find the solution. We'll use the average acceleration formula as our main tool.

Solving for the Initial Velocity

Alright, let's get down to the math! We'll start with the average acceleration formula we discussed earlier:

a_avg = (Δv) / (Δt)

We know that Δv (the change in velocity) is the final velocity (v) minus the initial velocity (v₀), and Δt (the change in time) is the final time (t) minus the initial time (t₀). So, we can rewrite the formula as:

a_avg = (v - v₀) / (t - t₀)

Now, let's plug in the values we know:

2 m/s² = (15 m/s - v₀) / (9 s - 4 s)

This simplifies to:

2 m/s² = (15 m/s - v₀) / 5 s

Our next step is to isolate the term with v₀. To do this, we'll multiply both sides of the equation by 5 s:

2 m/s² * 5 s = 15 m/s - v₀

This gives us:

10 m/s = 15 m/s - v₀

Now, we want to get v₀ by itself. We can add v₀ to both sides of the equation:

10 m/s + v₀ = 15 m/s

Finally, we subtract 10 m/s from both sides to solve for v₀:

v₀ = 15 m/s - 10 m/s

Therefore:

v₀ = 5 m/s

So, the initial velocity of the sled at 4 seconds was 5 m/s. Awesome! We've successfully calculated the initial velocity using the concepts of average acceleration and kinematics. Remember, the key is to understand the definitions, identify the given information, and carefully apply the formulas.

Real-World Applications and Why This Matters

You might be thinking,