Problem 10 - Applied Mathematics

Question

(a) A particle is moving in a straight line with a constant acceleration of \(2 \mathrm{~ms}^{-2}\) (b) A particle moving in a straight line with a constant acceleration has a velocity of \(2 \mathrm{~ms}^{-1}\) at one instant and a veloctty of \(8 \mathrm{~ms}^{-1}\) three seconds later.

Step-by-Step Solution

Answer

The constant acceleration is \(2 \; \mathrm{ms}^{-2}\) from both parts.

Step 1: Identify known quantities for Part (a)

For part (a), the constant acceleration given is \(a = 2 \; \mathrm{ms}^{-2}\). No additional information is provided, so we will note the acceleration and move to part (b) for further calculations.

Step 2: Identify known quantities for Part (b)

For part (b), we are given an initial velocity \(u = 2 \; \mathrm{ms}^{-1}\), a final velocity \(v = 8 \; \mathrm{ms}^{-1}\), and the time interval \(t = 3 \; \text{seconds}\).

Step 3: Use the first equation of motion for Part (b)

The first equation of motion is \(v = u + at\). Substituting the given values: \(8 = 2 + 3a\).

Step 4: Solve for acceleration in Part (b)

Rearrange the equation to solve for \(a\): \(8 - 2 = 3a \). This simplifies to \(6 = 3a\), hence \(a = 2 \; \mathrm{ms}^{-2}\).

Step 5: Compare with Part (a)

The calculated acceleration in part (b) matches the given acceleration in part (a), confirming the consistency in the problem statement.

Key Concepts

kinematics

Kinematics is a branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It's all about describing how objects move, using terms like displacement, velocity, and acceleration. Understanding kinematics is essential because it provides the foundation for analyzing and predicting the movement of objects.

equations of motion

In kinematics, there are several key equations of motion that help us describe the behavior of a moving object. These equations are particularly useful when dealing with constant acceleration. One such equation is
\ v = u + at \
Here, \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the constant acceleration, and \( t \) is the time interval over which the motion occurs. This equation lets us predict the velocity of an object at any given time, as we can see from the exercise: when an object’s acceleration and time are known, we can find how fast that object will be moving at a certain moment in the future.

velocity

Velocity is a vector quantity, meaning it has both magnitude and direction. It tells us how fast an object is moving and in which direction. In kinematics, we often focus on linear motion, so velocity in a straight line is central to many calculations.
For instance, in the given problem, an object first has a velocity of \(2 \; \mathrm{m/s}\) and later a velocity of \(8 \; \mathrm{m/s}\) three seconds later. This change in velocity, coupled with the time interval, allows us to determine the constant acceleration the object experiences using the previously described equations of motion.