A cheetah can run at a maximum speed
103 km/h and a gazelle can run at a maximum speed of 74.3 km/h.
If both animals are running at full speed,
with the gazelle 57.5 m ahead, how long before
the cheetah catches its prey? Answer in s.
The cheetah can maintain its maximum speed
for only 7.5 s.
What is the minimum distance the gazelle
must be ahead of the cheetah to have a chance
of escape? (After 7.5 s the speed of cheetah is
less than that of the gazelle.)
Answer in units of m.
I really dont understand this please help
distance = speed * time
when are the two distances equal? The cheetah has to run an extra 57.5m
To solve the first part of the problem, where we need to find the time it takes for the cheetah to catch the gazelle, we can set up an equation using the relative speed of the cheetah to the gazelle.
Since the cheetah is running faster, its relative speed will be the difference between the two speeds:
Relative speed = 103 km/h - 74.3 km/h = 28.7 km/h
Now, we need to convert the relative speed from km/h to m/s, since the given distance is in meters and the answer should be in seconds.
1 km/h = (1000 m)/(1 h) [Convert km to m]
1 h = 3600 s [Convert h to s]
So, 1 km/h = (1000 m)/(3600 s)
Using this conversion factor, we can convert the relative speed to m/s:
Relative speed = 28.7 km/h × (1000 m)/(3600 s) = 7.97222 m/s
Now, we can use the formula: time = distance/speed to find the time it takes for the cheetah to catch the gazelle.
Given that the gazelle is 57.5 m ahead, we can set up the equation:
time = 57.5 m / 7.97222 m/s
Simplifying, we find:
time = 7.2056 seconds
Therefore, it takes approximately 7.2056 seconds for the cheetah to catch the gazelle.
Moving on to the second part of the problem:
We are asked to find the minimum distance the gazelle must be ahead of the cheetah to have a chance of escape after 7.5 seconds.
In 7.5 seconds, the cheetah can cover a distance of (7.5 s) × (103 km/h) × (1000 m)/(3600 s) = 1,415.97222 m (using the conversion factors mentioned earlier).
To ensure that the gazelle can escape, it needs to be at least 1,415.97222 m ahead of the cheetah.
Therefore, the minimum distance the gazelle must be ahead of the cheetah is approximately 1,416 meters.
To solve this problem, we need to look at the speeds and distances of the cheetah and the gazelle. Let's break it down step by step:
Step 1: Convert the speeds to meters per second (m/s)
The cheetah's speed is 103 km/h. To convert it to m/s, we need to divide it by 3.6 (since there are 3.6 km in 1 hour).
So, the cheetah's speed is 103 km/h ÷ 3.6 = 28.6 m/s.
The gazelle's speed is 74.3 km/h. Again, we divide it by 3.6 to convert it to m/s.
So, the gazelle's speed is 74.3 km/h ÷ 3.6 = 20.6 m/s.
Step 2: Calculate the time it takes for the cheetah to catch up to the gazelle.
Let's assume the time it takes for the cheetah to catch up is t (in seconds).
During this time, the gazelle will be 57.5 m ahead of the cheetah.
We can use the equation distance = speed × time to calculate the distance traveled by both the cheetah and the gazelle during time t.
For the gazelle:
Distance gazelle = speed gazelle × time = 20.6 m/s × t
For the cheetah:
Distance cheetah = speed cheetah × time = 28.6 m/s × t
Since the gazelle starts 57.5 m ahead, we can set up the equation:
Distance gazelle - Distance cheetah = 57.5 m
Plugging in the expressions we derived:
20.6 m/s × t - 28.6 m/s × t = 57.5 m
Step 3: Solve the equation to find the time (t).
Combine like terms:
-8.0 m/s × t = 57.5 m
Divide both sides of the equation by -8.0 m/s:
t = 57.5 m / -8.0 m/s = -7.2 s
The negative value for time doesn't make sense in this context, so it means that the cheetah cannot catch up to the gazelle if it starts 57.5 m ahead. We need to adjust the distance so that the cheetah can have a chance of catching the gazelle.
Step 4: Calculate the minimum distance the gazelle must be ahead for the cheetah to have a chance of catching it.
Since the cheetah can maintain its maximum speed for only 7.5 seconds, we need to consider that when calculating the minimum distance.
Using the same approach as in Step 2, we can set up the equation:
Distance gazelle - Distance cheetah = 7.5 s × speed cheetah
Plugging in the expressions:
20.6 m/s × t - 28.6 m/s × t = 7.5 s × 28.6 m/s
Simplifying:
-8.0 m/s × t = 7.5 s × 28.6 m/s
Divide both sides of the equation by -8.0 m/s:
t = (7.5 s × 28.6 m/s) / -8.0 m/s
Simplifying:
t = -26.625 s
Again, the negative value doesn't make sense in this context, so we need to adjust the distance to give the gazelle a chance to escape.
Therefore, the minimum distance the gazelle must be ahead of the cheetah is the minimum distance required for the cheetah to catch the gazelle after 7.5 seconds. This distance is given by:
Distance gazelle = 7.5 s × speed cheetah = 7.5 s × 28.6 m/s
Calculating:
Distance gazelle = 7.5 s × 28.6 m/s = 214.5 m
Therefore, the minimum distance the gazelle must be ahead of the cheetah to have a chance of escape is 214.5 meters.