An object undergoes acceleration \(2.3 \hat{\imath}+3.6 \hat{\jmath} \mathrm{m} / \mathrm{s}^{2}\) for \(10 \mathrm{s}\). At the end of this time, its velocity is \(33 \hat{\imath}+15 \hat{\jmath} \mathrm{m} / \mathrm{s} .\) (a) What was its velocity at the beginning of the 10 -s interval? (b) By how much did its speed change? (c) By how much did its direction change? (d) Show that the speed change is not given by the magnitude of the acceleration multiplied by the time. Why not?

Understanding how to determine the initial velocity of an object in motion is a fundamental concept in kinematics. When you are presented with a scenario where you know the final velocity, the acceleration, and the time interval, you can work backwards to find the starting velocity. The equation used is a rearrangement of the equation for uniform acceleration:

\[\begin{equation} \vec{v}_{i} = \vec{v}_{f} - \vec{a}t \end{equation}\]

This allows us to subtract the product of acceleration and time from the final velocity. This subtraction isn't just simple arithmetic—it involves vector mathematics, since velocity and acceleration are vector quantities, meaning they have both magnitude and direction. Using this equation is a practical application of vector mathematics in physics, allowing us to analyze motion in multiple dimensions, as often encountered in real-world scenarios.

The concept of 'change in speed' refers to the difference in magnitude of an object's velocity over a period of time. It's important not to confuse this with acceleration, which takes into account direction as well as magnitude. To find the change in speed, you use the following formula:

\[\begin{equation} \Delta \text{speed} = |\vec{v}_{f}| - |\vec{v}_{i}| \end{equation}\]

where the vertical bars denote the magnitude (or speed) of the velocity vectors. This calculation involves taking the square root of the sum of the squares of the velocity components for both the initial and final velocities. Taking the magnitude of each vector results in scalars – numerical values without direction – which can then be subtracted to find the change in the object's speed regardless of the direction it's moving.

Vector mathematics is crucial in physics, especially when dealing with quantities like velocity and acceleration. A vector quantity includes both a magnitude and a direction. For instance, in the given exercise, velocity and acceleration are expressed in vector form, using the unit vectors \(\hat{\imath}\) and \(\hat{\jmath}\) to denote direction along the x and y axes respectively. To manipulate these vector quantities, we must carry out vector addition or subtraction. For example, when we calculate initial velocity, we subtract the acceleration vector multiplied by time from the final velocity vector, honoring both magnitude and direction in the respective calculations. Additionally, vector mathematics helps us determine the change in direction of an object by employing the dot product to calculate the angle between two vectors, providing a quantitative measure of direction change.

Acceleration is a vector quantity that represents how an object's velocity changes over time. It's defined as the rate of change of velocity and can result in an increase or decrease in the speed of an object, a change in its direction of motion, or both. In kinematics, acceleration is often constant, which simplifies calculations as shown in our exercise. However, it's important to understand that the magnitude of the acceleration vector does not directly give you the change in speed. This is because acceleration can impact the velocity's direction, not just its magnitude. While an object can have a high acceleration and a large change in direction, its speed might not change much, if at all - a concept students sometimes find counterintuitive. This is why multiplying the magnitude of the acceleration by the time elapsed gives us a different value from the actual speed change, which only concerns the magnitude and not the direction, as seen in the exercise.