Question

A car is traveling due west at $20.0\mathrm{m}/\mathrm{s}$. Find the velocity of the car after $37.00\mathrm{s}$ if its constant acceleration is $1.0\mathrm{m}/{\mathrm{s}}^2$ due east. Assume the acceleration remains constant. a) $17.0\mathrm{m}/\mathrm{s}$ west b) $17.0\mathrm{m}/\mathrm{s}$ east c) $23.0\mathrm{m}/\mathrm{s}$ west d) $23.0\mathrm{m}/\mathrm{s}$ east e) $11.0\mathrm{m}/\mathrm{s}$ south

Short answer

Answer: b) 17.0 m/s east

Step-by-step solution

Calculate the cumulated velocity due to acceleration

To compute the accumulated velocity due to the acceleration, use the formula: $V_{acc}=a\cdot t$ where $V_{acc}$ is the accumulated velocity, $a$ is the acceleration, and $t$ is the time duration. Given that, $a=1.0\mathrm{m}/{\mathrm{s}}^2$ and $t=37.00\mathrm{s}$, $V_{acc}=1.0\mathrm{m}/{\mathrm{s}}^2\cdot 37.00\mathrm{s}=37.0\mathrm{m}/\mathrm{s}$

Determine the final velocity

We know that the initial velocity ($V_{initial}$) is $20.0\mathrm{m}/\mathrm{s}$ west, and the accumulated velocity due to acceleration is $37.0\mathrm{m}/\mathrm{s}$. Since both the initial velocity and the accumulated velocity due to acceleration are in opposite directions (west and east), we subtract the accumulated velocity from the initial velocity to find the final velocity: $V_{final}=V_{initial}-V_{acc}$ $V_{final}=20.0\mathrm{m}/\mathrm{s}-37.0\mathrm{m}/\mathrm{s}$ $V_{final}=-17.0\mathrm{m}/\mathrm{s}$ Since the result is negative, it means the final velocity is in the opposite direction to the initial velocity, which is east. So, the final velocity of the car after $37.00\mathrm{s}$ is $17.0\mathrm{m}/\mathrm{s}$ east. The correct answer is option b) $17.0\mathrm{m}/\mathrm{s}$ east.

Key concepts

Kinematics

Kinematics is a branch of physics that deals with the motion of objects without considering the forces that cause this motion. It helps us understand and predict how objects move under different circumstances. Kinematic equations are essential tools used to describe the motion of objects along a straight line, usually expressed in terms of velocity, acceleration, and time. In the context of the problem with the car traveling west, kinematics helps us predict what happens when a constant acceleration is applied in the opposite direction.

The key kinematic equations allow you to connect different variables like initial velocity, final velocity, acceleration, and time. They make it easier to solve real-world problems involving motion. In summary, kinematics lays the groundwork for understanding any motion-related problems by offering a set of equations and concepts that model how objects move.

Velocity Calculation

Velocity is a vector quantity, which means it has both magnitude and direction. Unlike speed, which only measures how fast something is moving, velocity tells us how fast and in what direction an object is moving. Calculating velocity is pivotal for understanding motion, as it allows us to evaluate how an object’s position changes with time.

In the given exercise, we begin with an initial velocity of the car, given as $20.0\mathrm{m}/\mathrm{s}$ in the west direction. The problem statement asks us to determine the final velocity after a constant acceleration is applied. This involves calculating the change in velocity due to acceleration and then adjusting the initial velocity.

The formula used is:
$V_{final}=V_{initial}+V_{acc}$
but since directions differ, we effectively subtract:
$V_{final}=V_{initial}-V_{acc}$

By calculating properly, we find out not only the numeric value of the velocity but also its new direction, showing the total influence of the applied acceleration.

Constant Acceleration

Acceleration is defined as the rate of change of velocity over time, and when this rate is constant, it simplifies many calculations in physics. Constant acceleration means that an object's velocity changes by the same amount each second. This concept is highly significant because it makes predictions about how an object moves over time straightforward.

In problems like the one with the car, where acceleration is constant, we can use established kinematic formulas like:
$V_{acc}=a\times t$
Using this formula, we calculate how much the velocity changes. In our example, the car has a constant acceleration of $1.0\mathrm{m}/{\mathrm{s}}^2$ east, affecting its velocity over a period of $37.00\mathrm{s}$.

The simplicity of constant acceleration allows us to directly relate time, acceleration, and change in velocity using a simple multiplication, making prediction of future states feasible and quite manageable in physics.