A golfer rides in a golf cart at an average speed of 3.10 m/s for 26.0 s. She then gets out of the cart and starts walking at an average speed of 1.20 m/s. For how long (in seconds) must she walk if her average speed for the entire trip, riding and walking, is 2.30 m/s?
avg speed= total distance /time=(ride distance + walk distance*timewalking)/(26s+timewalking)
The Kentucky Derby is held at the Churchill Downs track in Louisville, Kentucky. The track is one and one-quarter miles in length. One of the most famous horses to win this event was Secretariat. In 1973 he set a Derby record that would be hard to beat. His average acceleration during the last four quarter-miles of the race was +0.0105 m/s2. His velocity at the start of the final mile (x = +1609 m) was about 16.58 m/s. The acceleration, although small, was very important to his victory. To assess its effect, determine the difference between the time he would have taken to run the final mile at a constant velocity of +16.58 m/s and the time he actually took. Although the track is oval in shape, assume it is straight for the purpose of this problem.
To find the time she must walk, we can use the concept of average speed.
The average speed formula is:
Average speed = Total distance / Total time
Let's break down the problem step by step:
1. The golfer rides in a golf cart at an average speed of 3.10 m/s for 26.0 s. This means she traveled a certain distance while riding the cart. We can find this distance using the formula:
Distance = Speed × Time
Distance while riding the cart = 3.10 m/s × 26.0 s
2. She then starts walking at an average speed of 1.20 m/s. We want to find the time she needs to walk. Let's assume she walks for t seconds.
Distance while walking = 1.20 m/s × t
3. The total distance of the trip is the sum of the distances traveled while riding the cart and walking:
Total distance = Distance while riding the cart + Distance while walking
Now comes the key step: the average speed for the entire trip is given as 2.30 m/s. We can set up the equation based on the average speed formula:
2.30 m/s = Total distance / Total time
4. Substituting the values we found earlier:
2.30 m/s = (Distance while riding the cart + Distance while walking) / (26.0 s + t)
Rewriting the equation without fractions:
2.30 m/s = (3.10 m/s × 26.0 s + 1.20 m/s × t) / (26.0 s + t)
We can now solve this equation to find the value of t, which represents the time she must walk.
Multiply both sides of the equation by (26.0 s + t):
2.30 m/s × (26.0 s + t) = 3.10 m/s × 26.0 s + 1.20 m/s × t
59.8 m + 2.30 m/s × t = 80.6 m + 1.20 m/s × t
Subtract 1.20 m/s × t from both sides:
59.8 m + 1.10 m/s × t = 80.6 m
Subtract 59.8 m from both sides:
1.10 m/s × t = 20.8 m
Divide both sides by 1.10 m/s to isolate t:
t = 20.8 m / 1.10 m/s
Calculating the result:
t ≈ 18.909 seconds
Therefore, she must walk for approximately 18.909 seconds to maintain an average speed of 2.30 m/s for the entire trip.