. The cheetah is the fastest running animal in the world. Cheetahs can accelerate to a speed of 20 m/s in 2.5 an can continue to accelerate to reach a stop speed of 29 m/s. Assume the acceleration is constant until the stop speed is reached and is zero thereafter.
(a) Express the cheetah's top speed in miles per hour.
(b) starting from a crouched position, how long does it take a cheetah to reach its top speed and how far does it travel in time?
(c) If a cheetah sees a warthog 120 m away how long will it take to reach lunch assuming the warthog does not move?
To solve these questions, we need to use the equations of motion that relate acceleration, time, and distance. The relevant equations are:
1. v = u + at
This equation relates the final velocity (v), initial velocity (u), acceleration (a), and time (t).
2. v^2 = u^2 + 2as
This equation relates the final velocity (v), initial velocity (u), acceleration (a), and displacement (s).
Now let's solve each part of the question step by step:
(a) Express the cheetah's top speed in miles per hour.
To convert meters per second (m/s) to miles per hour (mph), we'll use the following conversion factors:
1 mile = 1609.34 meters
1 hour = 3600 seconds
So, the conversion factor from m/s to mph is (1 mile / 1609.34 meters) * (3600 seconds / 1 hour).
Let's calculate the top speed in miles per hour:
Top speed = 29 m/s * (1 mile / 1609.34 meters) * (3600 seconds / 1 hour)
Therefore, the cheetah's top speed is approximately 64.8 mph.
(b) Starting from a crouched position, how long does it take a cheetah to reach its top speed, and how far does it travel in that time?
We know the initial velocity (u) is 0 m/s, final velocity (v) is 29 m/s, and the acceleration (a) is constant until the stop speed is reached. Let's calculate the time (t) required to reach the top speed:
v = u + at
29 m/s = 0 m/s + a * t
Here, a is the acceleration. Rearranging the equation:
t = v / a
t = 29 m/s / (20 m/s^2)
Therefore, it takes approximately 1.45 seconds for the cheetah to reach its top speed.
To calculate the distance traveled, we can use the equation:
s = ut + 0.5at^2
Since the initial velocity (u) is 0 m/s, the equation simplifies to:
s = 0.5at^2
s = 0.5 * 20 m/s^2 * (1.45 s)^2
Therefore, the cheetah travels approximately 21 meters in that time.
(c) If a cheetah sees a warthog 120 m away, how long will it take to reach lunch assuming the warthog does not move?
In this case, the cheetah starts from rest (0 m/s), and we need to find the time (t) required to cover a distance of 120 meters.
We can use the equation:
s = ut + 0.5at^2
Since acceleration (a) is constant until the stop speed is reached and zero thereafter, the equation simplifies to:
s = 0.5at^2
120 m = 0.5 * 20 m/s^2 * t^2
Simplifying further:
t^2 = (120 m * 2) / (20 m/s^2)
t^2 = 12 s^2
Taking the square root on both sides:
t = sqrt(12 s^2) = 3.46 seconds
Therefore, it will take approximately 3.46 seconds for the cheetah to reach the warthog.