The cheetah can reach a top speed of 114 km/h (71 mi/h). While chasing its prey in a short sprint, a cheetah starts from rest and runs 50 m in a straight line, reaching a final speed of 79 km/h.
Determine the cheetah's average acceleration during the short sprint.
m/s2
convert final speed to m/s
the average speed is half of the final speed
... because the start was from rest
divide the distance by the average speed
... this is the time
divide the final speed by the time
... this is the average acceleration
To determine the cheetah's average acceleration during the short sprint, we need to use the formula for acceleration:
acceleration (a) = (final velocity (v) - initial velocity (u)) / time (t)
In this case, we are given the final speed of the cheetah, but we don't have the time. However, we do have the distance the cheetah runs, which is 50m.
To find the time, we can use the formula:
distance (d) = initial velocity (u) * time (t) + (1/2) * acceleration (a) * time (t)^2
Since the initial velocity is zero (the cheetah starts from rest), the formula simplifies to:
distance (d) = (1/2) * acceleration (a) * time (t)^2
Plugging in the given values: d = 50m, u = 0, we can solve for time (t):
50m = (1/2) * a * t^2
Dividing both sides by (1/2) gives:
100m = a * t^2
Simplifying, we have:
t^2 = 100m / a
Now, we can use the final speed to find the time:
final velocity (v) = initial velocity (u) + acceleration (a) * time (t)
Plugging in the given values: v = 79 km/h = 79,000 m / 3600 s = 21.94 m/s, u = 0, and solving for time (t):
21.94 m/s = a * t
Substituting t^2 = 100m / a into the equation:
21.94 m/s = a * sqrt(100m / a)
Simplifying further:
21.94 m/s = sqrt(a * 100m)
Squaring both sides:
(21.94 m/s)^2 = a * 100m
483.3636 m^2/s^2 = a * 100m
Dividing by 100m:
4.8336 m/s^2 = a
Therefore, the cheetah's average acceleration during the short sprint is approximately 4.83 m/s2.