Question

After you apply the brakes, the acceleration of your car is in the opposite direction to its velocity. If the acceleration of your car remains constant, describe the motion of your car.

Short answer

Question: Describe the motion of a car when its acceleration is constant and in the opposite direction to its velocity after applying the brakes. Answer: After applying the brakes, the car experiences a constant deceleration which causes its velocity to decrease over time until it comes to a stop. This motion can be described using the equations of motion for constant acceleration, taking into account the negative value for deceleration.

Step-by-step solution

Understanding the given situation

We know that after applying the brakes, the acceleration of the car is in the opposite direction to its velocity. This means that the car is slowing down or decelerating. Since the acceleration is remaining constant, we can say that the deceleration is constant.

Visualizing the motion

To better understand the motion, imagine a car moving in a straight line with an initial velocity. Once the brakes are applied, the car starts to slow down at a constant rate until it eventually comes to a stop.

Describing the motion mathematically

Let's denote the initial velocity of the car as \(v_{0}\) and its constant deceleration as \(a\). In order to describe the motion of the car, we will use the equations of motion for constant acceleration (or deceleration, in this case): 1. \(v = v_{0} + at\): This equation gives us the velocity (\(v\)) of the car at any time \(t\). 2. \(s = v_{0}t + \frac{1}{2}at^2\): This equation gives us the distance (\(s\)) traveled by the car at any time \(t\). Since the car has a constant deceleration, the value of \(a\) will be negative. This will result in a decreasing velocity over time as described by the first equation. Eventually, the car will come to a stop when \(v = 0\).

Time and distance to come to a stop

To find out the time it takes for the car to stop and the distance it travels during this time, we can set \(v = 0\) in the first equation: \(0 = v_{0} + at\), and solve for time \(t\). Then, we can use this value of \(t\) in the second equation to find the distance the car traveled during this time. 1. \(t = \frac{-v_{0}}{a}\): Time it takes for the car to come to a stop. 2. \(s = v_{0}\frac{-v_{0}}{a} + \frac{1}{2}a\left(\frac{-v_{0}}{a}\right)^2\): Distance traveled by the car during this time. In conclusion, after applying the brakes, the car experiences a constant deceleration which causes its velocity to decrease until it comes to a stop. The motion of the car can be described using the equations of motion for constant acceleration (or deceleration).

Key concepts

Constant Acceleration

When we talk about constant acceleration, we're referring to a situation where an object's velocity changes at a steady rate over time. This is a basic concept that governs the motion of objects. In the scenario of a car braking, constant deceleration is the term used because the car's speed is steadily decreasing rather than increasing. Mathematically, constant acceleration (or deceleration) is represented by a fixed number in the acceleration variable 'a' in the equations of motion. It means that the object will speed up (for acceleration) or slow down (for deceleration) by the same amount each second. It's important to note that acceleration can be positive (increasing speed), zero (constant speed), or negative (decreasing speed), as in the case of our braking car. The idea is crucial because it simplifies the analysis of motion as we can reliably predict the future position or velocity of an object if we know its current state and its constant acceleration.

Motion Equations

The motion equations are a set of formulas that describe how an object moves under constant acceleration. There are usually four equations to characterize such motion, but in the exercise, two have been specifically used. The first equation,
\[v = v_{0} + at\]
is the velocity-time relationship, indicating how velocity 'v' changes over time 't' when starting from an initial velocity 'v₀' and undergoing a constant acceleration 'a'.

The second equation,
\[s = v_{0}t + \frac{1}{2}at^2\]
is the distance-time relationship, telling us the distance 's' traveled over time 't', starting with initial velocity 'v₀' and a constant acceleration 'a'. These equations are pivotal because they give us the ability to predict the final state of an object (such as its final velocity or position) from its initial conditions. When the acceleration is negative, as in deceleration, these equations still apply, but the acceleration value is substituted with a negative number to indicate the reduction in velocity.

Velocity-Time Relationship

The velocity-time relationship is fundamental in understanding motion. It describes how the velocity of an object changes with time. In graphical terms, for constant acceleration, this relationship is represented by a straight line when plotted on a velocity-time graph. The slope of this line is equal to the rate of acceleration. In the context of our car braking, this line would slope downwards, reflecting decreasing speed, and the slope would denote the constant deceleration. If you were to calculate the area under this line from the start to any point in time, it would give you the distance covered by the car in that time.

Understanding the velocity-time relationship allows us to analyze motion and force systems effectively. It also sets the foundation for more complex scenarios where acceleration is not constant but instead varies with time. For a constant deceleration, the plot simplifies the visualization of the car's motion from the moment brakes are applied until the vehicle comes to a complete stop, reinforcing the concepts discussed with actual data that can be graphically represented and easily interpreted.