Chapter 2: Q. 61 (page 62)

Ann and Carol are driving their cars along the same straight road. Carol is located at x = 2.4 mi at t = 0 h and drives at a steady 36 mph. Ann, who is traveling in the same direction, is located at x = 0.0 mi at t = 0.50 h and drives at a steady 50 mph.
a. At what time does Ann overtake Carol?
b. What is their position at this instant?
c. Draw a position-versus-time graph showing the motion of both Ann and Carol.

Short Answer

Expert verified

Ann's car would cross Carol's car after 2 hours at a distance of 74.4 m

Step by step solution

Step 1. Write the given information

Let the initial point of Carol is Y

The velocity of Carol is ${(v_{i})}_{c} = 36.0 m / h$

The position of Carol is role="math" localid="1648444538537" ${(x_{i})}_{c} = 2.4 m$

The time at this instant is role="math" localid="1648444715128"

{(t_{i})}_{c} = 0 h

Similarly,
The initial point of Ann is X

The velocity of Ann is ${(v_{i})}_{A} = 50 m / h$

The position of Ann is ${(x_{i})}_{A} = 0 m$
The time at this instant is ${(t_{i})}_{A} = 0.5 h$

Step 2. (a) To determine the time at which Ann will cross Carol

Let the point of crossing is Z. At this point let the time is $t$
The position of Carol at this point is ${(x_{f})}_{c}$

The position of Ann at this point is ${(x_{f})}_{A}$

Write the equation of motion at point Z for Carol

${(x_{f})}_{c} = {(x_{i})}_{c} + {(v_{i})}_{c} [t - {(t_{i})}_{c}]$
Insert the known variables in the above expression,

localid="1648445871907" ${(x_{f})}_{c} = 2.4 + 36 [t - 0] {(x_{f})}_{c} = 2.4 + 36 t . . . . . . (1)$

Similarly, write the equation of motion for Ann at point Z

${(x_{f})}_{A} = {(x_{i})}_{A} + {(v_{i})}_{A} [t - {(t_{i})}_{A}] {(x_{f})}_{A} = 0 + 50 [t - 0.5] {(x_{f})}_{A} = 50 t - 25 . . . . . . . (2)$

Compare the equations (1) and (2)
$2.4 + 36 t = 50 t - 25 27.4 = 14 t t = 2 h$
Thus, after 2 hours, Ann's car would cross Carol's car.

Step3. (b) To determine the position of crossing

The position of crossing is determined by substituting the value of $t$ in equation (1 ) or (2),

Consider equation (1)

${(x_{f})}_{c} = 2.4 + 36 (2) {(x_{f})}_{c} = 74.4 m$
Thus, Ann's car would cross Caol's car at 74.4 m.

To draw the position-time graph of Carol and Ann

The position-time graph of Carol and Ann's car is shown below. The red line shows the change in position of Carol's car and the blue line shows the change in position of Ann's Car. They both meet at the time of 2 hours at a distance of 74.4 meters.