91Ó°ÊÓ

Deceleration is a type of acceleration that causes an object to slow down. In our problem, the bicyclist is decelerating because their speed is decreasing over time. Deceleration is often represented with a negative value since it reduces the object's velocity.

To calculate deceleration, you can use the formula: \[ a = \frac{v_f - v_i}{t} \] where:

• a is the deceleration
• v_f is the final velocity (0 m/s in this case)
• v_i is the initial velocity
• t is the time over which the change occurs

In this exercise, the bicyclist decelerates from 8.33 m/s to 0 m/s in 45 seconds. Plugging in these values, we get: \[ a = \frac{0 \text{ m/s} - 8.33 \text{ m/s}}{45 \text{ s}} = -0.185 \text{ m/s}^2 \] This means the bicyclist's velocity decreases by 0.185 m/s every second.

To find the velocity after a specific amount of time, we use the formula: \[ v = v_i + a t \] where:

• v is the final velocity
• v_i is the initial velocity
• a is the acceleration (or deceleration)
• t is the time

In our example, we want to find the bicyclist's velocity after 20 seconds. Given:

• v_i = 8.33 m/s
• a = -0.185 m/s^2
• t = 20 s

we plug these into the formula: \[ v = 8.33 \text{ m/s} + (-0.185 \text{ m/s}^2 \times 20 \text{ s}) \] This simplifies to: \[ v = 8.33 \text{ m/s} - 3.7 \text{ m/s} = 4.63 \text{ m/s} \] So, after 20 seconds, the bicyclist is traveling at 4.63 m/s.

To calculate the distance traveled during deceleration, we use the formula: \[ s = v_i t + \frac{1}{2} a t^2 \] where:

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

For our problem, we need to find the distance traveled in the entire 45 seconds. Given:

• v_i = 8.33 m/s
• a = -0.185 m/s^2
• t = 45 s

we insert these into the formula: \[ s = 8.33 \text{ m/s} \times 45 \text{ s} + \frac{1}{2} (-0.185 \text{ m/s}^2) \times (45 \text{ s})^2 \] Breaking it down step-by-step, we get:

• First term: 8.33 m/s × 45 s = 374.85 m
• Second term: 0.5 × (-0.185 m/s^2) × (2025 s²) = -187.3125 m

Finally, adding these together: \[ s = 374.85 \text{ m} - 187.3125 \text{ m} = 187.54 \text{ m} \] This means the bicyclist traveled 187.54 meters during the 45 seconds it took to come to a complete stop.

Converting units is often a necessary step in solving physics problems, especially in kinematics. In this exercise, we need to convert the initial speed from km/hr to m/s.

To convert from kilometers per hour to meters per second, use the conversion factors:

• 1 kilometer = 1000 meters
• 1 hour = 3600 seconds

This gives us: \[ 30 \text{ km/hr} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ hr}}{3600 \text{ s}}= 30 \times \frac{1000}{3600} \] Simplifying this: \[ 30 \times \frac{1000}{3600} = 8.33 \text{ m/s} \] Converting units correctly ensures accuracy in your calculations and helps in comparing different quantities easily. By using the standard unit of meters per second (m/s), we can seamlessly integrate all other calculations in our kinematics problem.