91Ó°ÊÓ

Acceleration is a fundamental concept in kinematics that describes how quickly an object's velocity changes. When an object speeds up, slows down, or changes direction, it is experiencing acceleration. In our problem, the vehicle's acceleration is given as \(10.0 \mathrm{~m/s}^2\). This means that every second, the vehicle's velocity increases by \(10.0 \mathrm{~m/s}\). Acceleration can be positive (when speed increases) or negative (when speed decreases, also known as deceleration). Understanding acceleration helps us predict how fast something will be moving at any point in time during its motion.

Equations of motion are a set of mathematical formulas that describe the relationship between displacement, velocity, acceleration, and time. These equations allow us to predict the future position and velocity of an object if its acceleration and initial velocity are known.

The three main equations of motion are:

• \( v = u + at \)
• \( s = ut + \frac{1}{2} a t^2 \)
• \( v^2 = u^2 + 2as \)

In our exercise, we used the second equation \( s = ut + \frac{1}{2} a t^2 \) to find the distance since it directly relates to our given values, initial velocity \( u_i = 0 \), acceleration \( a \), and time \( t \). By understanding these equations, solving motion problems becomes systematic and straightforward.

The key to finding the distance traveled by the vehicle lies in using the right formula and plugging in the correct numbers. The distance formula derived from the equations of motion, \( s = ut + \frac{1}{2} a t^2 \), simplifies the process.

In our problem, the vehicle starts from rest, so the initial velocity \( u_i \) is \(0\). This means the distance formula becomes \( s = \frac{1}{2} a t^2 \). By substituting the given values for acceleration \( a = 10.0 \; \mathrm{m/s^2} \) and time \( t = 20.0 \; \mathrm{s} \) into the formula, you get:

• Calculate \( t^2 = 20^2 = 400 \)
• Multiply \( \frac{1}{2} \times 10.0 \times 400 \)
• The result is \( 2000 \; \mathrm{m} \)

Thus, the vehicle travels a distance of \( 2000 \; \mathrm{m} \) during the acceleration.

Velocity is a vector quantity that indicates the direction and speed of an object's movement. It is different from speed, which is a scalar quantity measuring how fast an object is moving regardless of direction.

In the context of acceleration, initial velocity (\( u_i \)) and final velocity (\( v \)) are crucial. Our exercise assumes an initial velocity of \(0\), as the vehicle starts from rest. As the vehicle accelerates, its velocity increases linearly over time. You can calculate the final velocity using \( v = u_i + at \). For our exercise, the calculation becomes:

• \( v = 0 + 10.0 \times 20 \)
• \( v = 200 \; \mathrm{m/s} \)

This means that after 20 seconds, the vehicle will be traveling at \(200 \; \mathrm{m/s}\). Knowing both velocity and acceleration helps determine overall motion dynamics.