Question

A car is moving at \(60 \mathrm{mi} / \mathrm{hr}(88 \mathrm{ft} / \mathrm{s})\) on a straight road when the driver steps on the brake pedal and begins decelerating at a constant rate of \(10 \mathrm{ft} / \mathrm{s}^{2}\) for 3 seconds. How far did the car go during this 3 -second interval?

Short answer

Answer: The car travelled 219 feet during the 3-second interval while decelerating.

Step-by-step solution

List the given information

The following information is provided: - Initial velocity (u): \(88 \ \text{ft/s}\) - Deceleration (a): \(-10 \ \text{ft/s}^2\) (Negative since it's deceleration) - Time (t): \(3 \ \text{s}\)

Use the distance formula with constant acceleration

The formula for distance (s) travelled with constant acceleration (or deceleration) is: \(s = ut + \frac{1}{2}at^2\)

Plug in the given values and solve for s

Substitute the given values into the formula: \(s = 88(3) + \frac{1}{2}(-10)(3)^2\) \(s = 264 - \frac{1}{2}(90)\) \(s = 264 - 45\)

Calculate the distance

Calculate the distance travelled: \(s = 219 \ \text{ft}\) The car travelled 219 feet during the 3-second interval while decelerating.

Key concepts

Constant Acceleration

When studying motion, one of the fundamental concepts is constant acceleration. This is the case when the velocity of an object changes at a steady rate over time. In kinematics, which is the branch of physics that deals with objects in motion, we use formulas to calculate various aspects of this motion.

For a body moving with constant acceleration, the velocity at any instance can be found using the formula: \( v = u + at \), where \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time elapsed. However, if we're interested in the distance traveled, we use a different equation, which we'll discuss in the distance formula section. The concept of constant acceleration is crucial in explaining various everyday phenomena, such as the motion of a car increasing or decreasing its speed at a steady rate.

Deceleration

Deceleration, sometimes referred to as negative acceleration, occurs when an object is slowing down. The rate at which an object decelerates is measured in units such as \( \text{ft/s}^2 \) or \( \text{m/s}^2 \) and indicates how quickly the speed decreases per second.

It's important to note that deceleration is mathematically treated as acceleration with a negative value. This is because acceleration is a vector quantity and can therefore have direction - in the case of deceleration, the direction is opposite to that of the motion. When solving problems involving deceleration, remembering to use a negative value for acceleration helps avoid confusion and yields the correct results. For instance, if a car is moving at a speed of \(88 \text{ ft/s}\) and begins to decelerate at \(10 \text{ ft/s}^2\), you would use \(a = -10 \text{ ft/s}^2\) in your calculations.

Distance Formula

The distance formula in kinematics is a mathematical relation that allows us to calculate the distance traveled by an object moving with constant acceleration over a certain amount of time. The formula is written as: \( s = ut + \frac{1}{2}at^2 \), where \( s \) represents the distance traveled, \( u \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time.

This equation is derived from integrating the velocity with respect to time and gives us a clear picture of how these quantities are interrelated. The formula essentially states that the distance is equal to the initial velocity multiplied by time added to half of the acceleration multiplied by the square of time. It's a powerful tool for solving problems in classical mechanics and helps students understand the motion of objects in a more comprehensive manner.

Initial Velocity

Initial velocity is the speed and direction with which an object begins its motion. In the case of linear motion, we can simply refer to it as the speed at which an object starts moving before any forces (like acceleration due to gravity or applied brakes in a car) act upon it.

The significance of initial velocity lies in its role in kinematic equations, as it is often a starting point for calculations. Knowing the initial velocity allows us to predict the future position and speed of the object using various formulas, provided other factors like acceleration and time are known. For example, if a car has an initial velocity of \(88 \text{ ft/s}\) and starts to decelerate, its velocity at any later time can be determined based on that initial value. Initial velocity is denoted by \( u \) in equations and is a key variable in understanding motion.