Examkin

Question

Difficulty: MediumRate, Time, and Distance Problems

A delivery van leaves a warehouse traveling along a straight highway at a constant speed of 4545 miles per hour. Exactly 4040 minutes later, a courier motorcycle leaves the same warehouse along the exact same route, traveling at a constant speed of 6060 miles per hour. How many hours after the courier motorcycle departs will it overtake the delivery van?

Answer: 2 hours

Answer

The courier motorcycle will overtake the delivery van 2 hours after the motorcycle departs.
The delivery van travels for 23\frac{2}{3} hour before the motorcycle begins moving, establishing a initial lead of 45×23=3045 \times \frac{2}{3} = 30 miles. Since both vehicles travel in the same direction, the motorcycle gains on the van at a rate equal to the difference of their speeds, 60−45=1560 - 45 = 15 miles per hour. To cover the 30-mile gap, it takes 3015=2\frac{30}{15} = 2 hours after the motorcycle departs.

Step-by-Step Solution

1
Convert the van's head-start time from minutes to hours.
Head-start time is 4060=23\frac{40}{60} = \frac{2}{3} hours.
Speed is given in miles per hour, so time units must be converted to hours for consistency.
2
Calculate the lead distance gained by the van before the motorcycle starts moving.
Lead distance = 45×23=3045 \times \frac{2}{3} = 30 miles.
Distance equals rate multiplied by time (d=r⋅td = r \cdot t).
3
Calculate the relative speed at which the motorcycle gains on the van.
Relative speed = 60−45=1560 - 45 = 15 miles per hour.
When two objects move in the same direction, the rate at which the distance between them decreases is the difference between their individual speeds.
4
Determine the time needed for the motorcycle to eliminate the 30-mile gap.
Time = 3015=2\frac{30}{15} = 2 hours.
Time equals distance divided by relative speed (t=drrelt = \frac{d}{r_{\text{rel}}}).

Key Concept

Catch-up scenarios and relative speed in same-direction movement.

Alternative Method

Equate the distance expressions for both vehicles at the moment of overtaking. Let tt represent the motorcycle's travel time in hours. The van's travel time is t+23t + \frac{2}{3} hours. Since both travel equal distances from the warehouse: 60t=45(t+23)  ⟹  60t=45t+30  ⟹  15t=30  ⟹  t=260t = 45\left(t + \frac{2}{3}\right) \implies 60t = 45t + 30 \implies 15t = 30 \implies t = 2 hours.
Estimated Time:1m 30s
Rate this question

Topics

Rate, Time, and Distance Problems

More questions from this topic