How fast is a car moving if it starts from rest and accelerates at a rate of 3.5 m/s² for 250 m?
Understand the Problem
The question is asking for the final speed of a car that starts from rest and accelerates at a rate of 3.5 m/s² over a distance of 250 meters. We can use the kinematic equation for uniformly accelerated motion to solve this problem.
Answer
The car reaches a speed of $41.8 \, \text{m/s}$.
Answer for screen readers
The final speed of the car is approximately $41.8 , \text{m/s}$.
Steps to Solve
- Identify the kinematic equation to use
We will use the equation for final velocity given constant acceleration:
$$ v^2 = u^2 + 2as $$
where:
- $v$ = final velocity
- $u$ = initial velocity (0 m/s, since the car starts from rest)
- $a$ = acceleration (3.5 m/s²)
- $s$ = distance (250 m)
- Substitute known values into the equation
Since the initial velocity $u$ is 0, the equation simplifies to:
$$ v^2 = 0 + 2 \cdot 3.5 \cdot 250 $$
- Calculate the right side of the equation
Now, compute the value:
$$ v^2 = 2 \cdot 3.5 \cdot 250 $$
This gives:
$$ v^2 = 1750 $$
- Take the square root to find final velocity
To find $v$, take the square root of both sides:
$$ v = \sqrt{1750} $$
- Calculate the final value
Using a calculator:
$$ v \approx 41.8 , \text{m/s} $$
The final speed of the car is approximately $41.8 , \text{m/s}$.
More Information
This result indicates that if a car accelerates at a constant rate of 3.5 m/s² from rest for 250 meters, it will reach a speed of 41.8 m/s. Kinematic equations are very useful for solving problems involving motion under constant acceleration.
Tips
- Forgetting to convert units or not using consistent units throughout the calculation. Always ensure acceleration is in m/s² and distance in meters.
- Misapplying the formula by omitting the initial velocity or not correctly identifying the variables.
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