91Ó°ÊÓ

Acceleration is a fundamental concept in kinematics, which describes how an object's velocity changes over time. It measures the rate of change of velocity per unit of time. In this exercise, we have an acceleration of \(+1.2 \, \mathrm{cm/s^2}\), which means every second, the velocity of the object increases by \(+1.2 \, \mathrm{cm/s}\).
Understanding acceleration helps predict how fast or slow an object's speed is changing. When the acceleration is positive, it means the object is speeding up in the direction of motion. Conversely, a negative acceleration indicates the object is slowing down.
For example:

• If the velocity of an object is negative, like \(-3.4 \, \mathrm{cm/s}\), a positive acceleration decreases the negativity (makes it less negative or more positive over time).
• Over time, this change can be calculated using the kinematics equations, allowing us to know the velocity at different moments.

Velocity calculation involves determining the current speed and direction of an object in motion. It is derived from knowing the object's initial velocity and how that velocity changes over time due to acceleration.
In our exercise, we start with an initial velocity of \(-3.4 \, \mathrm{cm/s}\) at \(t = 4.0 \, \mathrm{s}\). To calculate velocity at different times like \(t = 1.0 \, \mathrm{s}\) and \(t = 6.0 \, \mathrm{s}\), we must first determine how far these are from our initial time.To do this:

• Calculate time intervals: For \(t = 1.0 \, \mathrm{s}\), the interval is \(-3.0 \, \mathrm{s}\) and for \(t = 6.0 \, \mathrm{s}\), the interval is \(+2.0 \, \mathrm{s}\).
• Apply the kinematics equation by plugging these intervals into the formula \(v = v_0 + at\) to find the new velocities.

By understanding these intervals and how they relate to acceleration, we determine the velocities: \(-7.0 \, \mathrm{cm/s}\) at \(t = 1.0 \, \mathrm{s}\) and \(-1.0 \, \mathrm{cm/s}\) at \(t = 6.0 \, \mathrm{s}\). This shows how the object changes speed and direction over time.

The kinematics equation is a simple yet powerful tool for predicting an object's future velocity. It is based on the relationship between initial velocity, constant acceleration, and time. The formula used is \(v = v_0 + at\), where:

• \(v\) is the final velocity you're trying to find.
• \(v_0\) represents the initial velocity of the object.
• \(a\) is the acceleration, which remains constant here.
• \(t\) is the time elapsed since the initial observation.

Given the provided initial conditions—velocity at \(t = 4.0 \, \mathrm{s}\) and the object's acceleration—you can find how the velocity changes at different times by plugging numbers into this equation.
This method allows us to accurately predict velocities, like \(-7.0 \, \mathrm{cm/s}\) and \(-1.0 \, \mathrm{cm/s}\) for the object at \(t = 1.0 \, \mathrm{s}\) and \(t = 6.0 \, \mathrm{s}\) respectively. This equation is especially useful because it requires only straightforward arithmetic to solve complex motion problems, making it accessible for students learning kinematics.