Problem 81 - College Physics

Question

An observer by the side of a straight, level, northsouth road watches a car (A) moving south at a rate of \(75 \mathrm{~km} / \mathrm{h}\). A driver in another car (B) going north at \(50 \mathrm{~km} / \mathrm{h}\) also observes car \(\mathrm{A}\). (a) What is car A's velocity as observed from car B? (Take north to be positive.) (b) If the roadside observer sees car A brake to a stop in \(6.0 \mathrm{~s}\) what constant acceleration would be measured? (c) What constant acceleration would the driver in car B measure for the braking car A?

Step-by-Step Solution

Answer

(a) -125 km/h, (b) 3.47 m/s², (c) 3.47 m/s².

Step 1: Understanding Relative Velocity

To find the velocity of car A as observed from car B, we need to calculate the relative velocity of car A with respect to car B. This is done using the formula: \( v_{AB} = v_A - v_B \). Both velocities are given with respect to a stationary observer on the roadside.

Step 2: Calculate Individual Velocities

The velocity of car A \( v_A \) is \(-75 \text{ km/h}\) because it moves south. For car B, the velocity \( v_B \) is \(50 \text{ km/h}\) because it moves north. The sign indicates direction with north positive and south negative.

Step 3: Calculate Relative Velocity of A with Respect to B

Substitute the velocities into the relative velocity formula: \( v_{AB} = (-75) - 50 = -125 \text{ km/h} \). This means car A appears to move at \(-125 \text{ km/h}\) as seen from car B.

Step 4: Understanding Acceleration Calculation

Acceleration is calculated as the change in velocity over time. Since car A stops, its final velocity \( v_f = 0 \). Initial velocity \( v_i = -75 \text{ km/h} \). Convert to m/s: \(-75 \text{ km/h} = -20.83 \text{ m/s}\). Time \( t = 6.0 \text{ s} \).

Step 5: Calculate Acceleration for Observer

Use the formula \( a = \frac{v_f - v_i}{t} \) to find the acceleration measured by the roadside observer: \( a = \frac{0 - (-20.83)}{6.0} \approx 3.47 \text{ m/s}^2 \).

Step 6: Relative Acceleration from Car B’s Perspective

The acceleration experienced by car B for car A is the same as the roadside observer since acceleration is invariant under relative motion (for constant motion observers), thus remains \( a = 3.47 \text{ m/s}^2 \).

Key Concepts

Constant Acceleration

Constant acceleration describes a steady change in velocity over time. When a vehicle moves with constant acceleration, the rate at which its speed changes remains the same. In the given problem, car A undergoes constant acceleration when it brakes to stop. This means car A's velocity decreases at a consistent rate until it reaches zero.

We calculate constant acceleration using the formula: \[ a = \frac{v_f - v_i}{t} \] where:

• \(v_f\) is the final velocity
• \(v_i\) is the initial velocity
• \(t\) is the time period over which the change occurs

In car A's case, it goes from \(-20.83 \text{ m/s}\) to 0 in 6 seconds. Plugging these values into the formula gives an acceleration of approximately \(3.47 \text{ m/s}^2\).

The critical point to understand here is that even though car A's speed changes, the rate of this change—its acceleration—remains constant as it brakes.

Velocity Calculation

Velocity calculation involves determining the rate of change of position in a specific direction. It’s important to note both the magnitude and direction when discussing velocity, as it is a vector quantity. In the exercise, you’re asked to calculate the velocity of car A as observed by someone in car B.

To do this, we use the relative velocity formula, which helps find how fast one object is moving relative to another. The formula is: \[ v_{AB} = v_A - v_B \] Here,

• \(v_{A}\) is the velocity of car A
• \(v_{B}\) is the velocity of car B

Since both cars are moving in opposite directions, with north being positive, car A's velocity (southward) is \(-75 \text{ km/h}\) and car B's (northward) is \(50 \text{ km/h}\). When calculating \(v_{AB}\), it results in \(-125 \text{ km/h}\), meaning car A seems to move faster backward from car B's viewpoint.

Observer's Perspective

From an observer's perspective, understanding how motion appears depends greatly on their frame of reference. In the problem, there are two observers: one on the roadside and another in car B.

For the roadside observer, car A's deceleration is straightforward. They see it slowing down at a constant rate until it halts. The acceleration they observe is a direct calculation from how quickly car A's speed changes over the 6-second interval.

In contrast, the driver in car B experiences a more complex scenario. Not only do they perceive car A as moving in the opposite direction, but any changes in speed also consider car B's own velocity. However, when car A brakes, its acceleration magnitude remains the same from both perspectives because acceleration is constant irrespective of the observer’s motion, provided that both observers travel at uniform speeds. This invariance means that both observers measure the same acceleration value for car A.