You are driving along the street at the speed limit (35mph) and 50 meters before reaching a traffic light you notice it becoming yellow. You accelerate to make the traffic light within the 3 seconds it takes for it to turn red. What is your speed as you cross the intersection? Assume that the acceleration is constant and that there is no air resistance.

To determine your speed as you cross the intersection after accelerating, we can break down the problem into a few logical steps. We know your initial speed, the distance to the traffic light, the time until it turns red, and that your acceleration is constant. Let's work through the calculations together.

Initial Information

Here’s what we have:

• Initial speed (v₀): 35 mph
• Distance to the traffic light (d): 50 meters
• Time until the light turns red (t): 3 seconds

Convert Speed to Meters per Second

First, we need to convert your initial speed from miles per hour to meters per second for easier calculations. The conversion factor is:

1 mph ≈ 0.44704 m/s

So, 35 mph is:

35 mph × 0.44704 m/s ≈ 15.65 m/s

Using Kinematic Equations

Next, we can use the kinematic equation that relates initial velocity, acceleration, time, and final velocity:

v = v₀ + at

Where:

• v is the final velocity (what we want to find)
• v₀ is the initial velocity (15.65 m/s)
• a is the acceleration
• t is the time (3 seconds)

Finding Acceleration

Before we can find the final speed, we need to determine the acceleration. We can use another kinematic equation that relates distance, initial velocity, time, and acceleration:

d = v₀t + 0.5at²

Plugging in the values we have:

50 m = (15.65 m/s)(3 s) + 0.5a(3 s)²

Calculating the first part:

50 m = 46.95 m + 4.5a

Now, rearranging the equation to solve for acceleration (a):

50 m - 46.95 m = 4.5a

3.05 m = 4.5a

a ≈ 0.678 m/s²

Calculating Final Speed

Now that we have the acceleration, we can find the final speed using the first kinematic equation:

v = v₀ + at

v = 15.65 m/s + (0.678 m/s²)(3 s)

v = 15.65 m/s + 2.034 m/s

v ≈ 17.684 m/s

Final Conversion to mph

To express the final speed back in miles per hour:

17.684 m/s ÷ 0.44704 m/s ≈ 39.5 mph

Summary

As you cross the intersection, your speed would be approximately 39.5 mph. This demonstrates how acceleration over a short distance and time can significantly increase your speed, even when starting at the speed limit. Always remember to consider safety and traffic laws when driving!