Question

A particle starts from rest at \(x=0.00\) and moves for \(20.0 \mathrm{~s}\) with an acceleration of \(+2.00 \mathrm{~cm} / \mathrm{s}^{2} .\) For the next \(40.0 \mathrm{~s},\) the acceleration of the particle is \(-4.00 \mathrm{~cm} / \mathrm{s}^{2} .\) What is the position of the particle at the end of this motion?

Short answer

Answer: The final position of the particle is 1200 centimeters to the left of the starting position, or at \(x = -1200.0 \mathrm{~cm}\).

Step-by-step solution

Find position after first acceleration

The particle starts with an acceleration of \(+2.00 \mathrm{~cm} / \mathrm{s}^{2}\) for 20 seconds. First, we use the equation \(x=x_0+v_0t+\frac{1}{2}at^2\): \(x = 0.00 + 0 \cdot 20.0 + \frac{1}{2}(2.00)(20.0)^2\) \(x = 0 + 0 + 400 = 400.0 \mathrm{~cm}\). The position after the first acceleration is \(400.0 \mathrm{~cm}\).

Calculate final velocity after first acceleration

To find the initial velocity for the second part of the motion, we must calculate the final velocity after the first acceleration using the equation \(v = v_0 + at\): \(v = 0 + (2.00)(20.0) = 40.0 \mathrm{~cm/s}\).

Find position after second acceleration

Now the particle has an initial velocity of \(40.0 \mathrm{~cm/s}\) and is decelerated at \(-4.00 \mathrm{~cm} / \mathrm{s}^{2}\) for 40.0 seconds. We can use the earlier position equation to find the position after the second acceleration period: \(x = 400.0 + (40.0)(40.0) + \frac{1}{2}(-4.00)(40.0)^2\) \(x = 400.0 + 1600 - 3200 = -1200.0 \mathrm{~cm}\).

Calculate final position

The final position of the particle is 1200 centimeters to the left of the starting position (since it's negative), or at \(x = -1200.0 \mathrm{~cm}.\)

Key concepts

Uniformly Accelerated Motion

Understanding uniformly accelerated motion is key to analyzing the movement of objects in physics. It refers to a scenario where an object's velocity changes at a constant rate, i.e., the acceleration is uniform. This situation is ideal for studying because it simplifies calculations and can be easily predicted mathematically.

Since acceleration is defined as the change in velocity per unit time, a constant acceleration means the object’s speed is increasing or decreasing by the same amount each second. This is reflected in the example given, where a particle starts from rest and then moves with a defined acceleration. The first part of the example, with an acceleration of \( +2.00 \mathrm{~cm} / \mathrm{s}^{2} \), demonstrates this principle perfectly, showing how the particle's velocity increases gradually over time.

Final Velocity Calculation

Calculating the final velocity of an object under uniformly accelerated motion is a fundamental aspect of kinematics. Using the equation \( v = v_{0} + at \), where \( v \) is the final velocity, \( v_{0} \) is the initial velocity, \( a \) is the acceleration and \( t \) is the time, we can determine how fast an object will be moving after a certain period of acceleration.

In our example, the initial velocity \( v_{0} \) is zero because the particle starts from rest, and after the application of an acceleration \( a \) of \( 2.00 \mathrm{~cm} / \mathrm{s}^{2} \) over a time period \( t \) of 20 seconds, the final velocity achieved can be calculated. This step is crucial, as it not only indicates the speed at that instant but also serves as the initial velocity for the subsequent motion if the acceleration changes, as seen when the particle transitions to a decelerative phase.

Displacement with Constant Acceleration

The displacement of an object can tell us how far it has moved from its initial position. When dealing with constant acceleration, we can apply the kinematic equation \( x = x_{0} + v_{0}t + \frac{1}{2}at^{2} \) to find this displacement. Here, \( x \) is the final position, \( x_{0} \) is the initial position, \( v_{0} \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time.

This equation is derived from integrating the acceleration function to get velocity and integrating once more to get displacement. It is elegantly suited to determine how far the object travels, especially when the object starts from rest or has an initial velocity. In the given exercise, the equation allows us to compute the progression of the particle's journey, taking into account the change in acceleration mid-motion, which is particularly insightful for understanding the combination of two different phases of accelerated motion.