Problem 83 - Calculus and its applications

Question

A bicyclist decelerates at a constant rate from \(30 \mathrm{~km} / \mathrm{hr}\) to a complete stop in \(45 \mathrm{sec}\) a) How fast is the bicyclist traveling after 20 sec? b) How far has the bicyclist traveled after \(45 \mathrm{sec} ?\)

Step-by-Step Solution

Answer

a) 4.63 m/s after 20 sec. b) 187.54 m after 45 sec.

Step 1: Convert Initial Speed to m/s

The initial speed is given as \(30 \text{ km/hr}\). To convert this into m/s, we use the conversion factor \(1 \text{ km/hr} = \frac{1}{3.6} \text{ m/s}\). Hence, \(30 \text{ km/hr} = \frac{30}{3.6} \text{ m/s} \approx 8.33 \text{ m/s}\).

Step 2: Calculate Deceleration

The bicyclist comes to a stop in \(45 \text{ sec}\). The final velocity \(v_f\) is \(0 \text{ m/s}\), and the initial velocity \(v_i\) is \(8.33 \text{ m/s}\). Using the formula for constant acceleration, \(a = \frac{v_f - v_i}{t}\), the deceleration is \(a = \frac{0 - 8.33}{45} \approx -0.185 \text{ m/s}^2\).

Step 3: Calculate Velocity after 20 Seconds

Using the formula \(v = v_i + at\), where \(a = -0.185 \text{ m/s}^2\) and \(t = 20 \text{ sec}\), we find \(v = 8.33 + (-0.185 \times 20) = 8.33 - 3.7 = 4.63 \text{ m/s}\).

Step 4: Calculate Distance Traveled in 45 Seconds

To find the distance, use the formula \(s = v_i t + \frac{1}{2}at^2\). Substituting for \(v_i = 8.33 \text{ m/s}\), \(a = -0.185 \text{ m/s}^2\), and \(t = 45 \text{ sec}\), the distance is \(s = 8.33 \times 45 + \frac{1}{2} \times (-0.185) \times 45^2\). Thus, \(s = 374.85 - 187.3125 = 187.54 \text{ m}\).

Key Concepts

Deceleration

Deceleration is a term used to describe the process of slowing down or reducing speed. In physics, it is considered a negative acceleration. Unlike acceleration which speeds things up, deceleration does the opposite. To calculate deceleration, we use the formula:

\[ a = \frac{v_f - v_i}{t} \]
where:

• \( a \) is the acceleration (or deceleration in this case),
• \( v_f \) is the final velocity,
• \( v_i \) is the initial velocity,
• \( t \) is the time taken to slow down.

This formula helps in determining how quickly a vehicle or object loses speed over a certain period. When the bicyclist decelerates from an initial velocity of 8.33 m/s to a stop in 45 seconds, it results in a deceleration of \(-0.185\) m/s². This value represents how much speed is lost every second during the slowing down phase.

Conversion of Units

Conversion of units is essential in physics when different measurement systems are used. It ensures calculations are consistent and accurate. In this exercise, the initial speed is given in kilometers per hour (km/hr) but needs to be converted to meters per second (m/s) because our equations typically use base SI units. The conversion factor used is:

1 km/hr = \( \frac{1}{3.6} \) m/s
By applying this factor, the initial speed of 30 km/hr was converted to 8.33 m/s. Understanding these conversions helps to seamlessly switch between different units of measure in problem-solving scenarios.

Constant Acceleration

Constant acceleration, also known as uniform acceleration, occurs when an object's velocity changes at a steady rate over time. In the context of this problem, the bicyclist's speed decreases at a constant rate until it stops. The uniform rate of deceleration makes it easier to predict outcomes, like remaining velocity at certain times or total stopping distance.

The formula used here is:
\[ v = v_i + at \]
where:

• \( v \) is the velocity at time \( t \),
• \( v_i \) is the initial velocity,
• \( a \) is the constant acceleration,
• \( t \) is the time.

These principles allow us to address part (a) of the exercise, where we calculate the speed of the bicyclist after 20 seconds during deceleration.

Distance Calculation

Distance calculation in kinematics involves determining how far an object has traveled over a period of time when undergoing acceleration or deceleration. For this bicyclist problem, the distance traveled during the entire time of deceleration is calculated using the equation:

\[ s = v_i t + \frac{1}{2} a t^2 \]
This formula considers both the initial speed and the influence of the acceleration over time.

• \( s \) is the distance traveled,
• \( v_i \) is the initial velocity,
• \( a \) is the acceleration (or deceleration),
• \( t \) is the time.

In this scenario, after inputting the values, the total distance the bicyclist covers in 45 seconds is carefully calculated as 187.54 meters. This formula helps to break down the journey into understandable components, adding clarity to the outcome.