Question

Your friend's car starts from rest and travels $0.500\mathrm{km}$ in $10.0\mathrm{s}$. What is the magnitude of the constant acceleration required to do this?

Short answer

Answer: The magnitude of the constant acceleration required for the car to travel 0.500 km in 10.0 seconds is $10\frac{m}{s^2}$.

Step-by-step solution

Write the given data

The car starts from rest, so its initial velocity $v_0=0$. The distance traveled, $d=0.500km$ which needs to be converted to meters: $d=500m$. The time taken, $t=10.0s$.

Use the equation for distance traveled under constant acceleration

We will use the equation for the distance traveled under constant acceleration: $d=v_{0}t+\frac{1}{2}a{t}^2$ Since the initial velocity is 0, we can simplify the equation to: $d=\frac{1}{2}a{t}^2$ Now, plug in the given values: $500m=\frac{1}{2}a(10.0s{)}^2$

Solve for acceleration, a

First, calculate the value inside the parentheses: $(10.0s{)}^2=100s^2$ Now, multiply both sides by 2: $1000m=a(100s^2)$ Finally, divide both sides by 100 to find the acceleration: $a=\frac{1000m}{100s^2}=10\frac{m}{s^2}$ The magnitude of the constant acceleration required for the car to travel 0.500 km in 10.0 s is $10\frac{m}{s^2}$.

Key concepts

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without taking into account the forces that cause the motion. It's about describing how objects move—think speed, velocity, and acceleration—in space and time.

Understanding kinematics is crucial for solving many physics problems, particularly those involving constant acceleration. As in the exercise, we're often interested in determining the displacement (change in position), velocity, and acceleration of objects over time. In the given problem, we were tasked with calculating the car's acceleration, using its displacement over a set period.

Equations of Motion

Equations of motion are mathematical formulas that describe the kinematic behavior of moving objects. There are a few key equations that are commonly used when an object is accelerating at a constant rate:

• The first equation connects velocity, acceleration, and time: $v=v_0+at$.
• The second equation associates displacement, initial velocity, and time: $d=v_{0}t+\frac{1}{2}a{t}^2$.
• The third equation links displacement, velocity, and acceleration: $v^2=v_0^2+2ad$.

These equations are often referred to as the 'SUVAT' equations (from S = displacement, U = initial velocity, V = final velocity, A = acceleration, and T = time). By choosing the right equation based on the given and required variables, solving kinematics problems becomes more straightforward.

Acceleration Calculation

Acceleration is the rate at which an object's velocity changes over time. If this rate is constant, we say the object has constant acceleration. The formula to calculate constant acceleration when you know the displacement and time, as seen in our exercise example, is derived from the second SUVAT equation:$d=\frac{1}{2}a{t}^2$.

From this formula, we can solve for acceleration by rearranging the terms: $a=\frac{2d}{t^2}$. It's crucial to plug in the correct units—usually meters for displacement and seconds for time—to get the acceleration in meters per second squared ($\frac{m}{s^2}$).

Physics Problem Solving

Physics problem solving is a systematic process that involves understanding the problem, identifying the relevant physical principles, choosing the appropriate mathematical tools, and carrying out the necessary calculations. Here's a simplified approach to solving physics problems:

1. Read the problem carefully: Understand what is given and what needs to be found. In our example, we identified that the car's initial velocity, distance traveled, and time were given, and we needed to find the acceleration.
2. Sketch the situation: Sometimes a simple diagram can help visualize the problem.
3. Select the right formula: Based on the known and unknown variables, choose the equation that fits best.
4. Plug in the known values: Substitute the numbers into the formula, with attention to units.
5. Solve for the unknown: Rearrange the equation if necessary and perform the calculations to find the answer.

Following these steps can lead to successful problem-solving in physics, especially in kinematics and other areas.