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 49 m in a straight line, reaching a final speed of 71 km/h.
(a) Determine the cheetah's average acceleration during the short sprint.
m/s2
(b) Find its displacement at
t = 3.0 s.
(Assume the cheetah maintains a constant acceleration throughout the sprint.)
m
To find the cheetah's average acceleration during the short sprint, we can use the equation:
Average acceleration = (change in velocity) / (time)
In this case, the change in velocity is the final velocity minus the initial velocity. Given that the cheetah starts from rest, the initial velocity is 0 km/h. The final velocity is given as 71 km/h. To convert these velocities to m/s, we can use the conversion factor: 1 km/h = 0.2778 m/s.
Initial velocity = 0 km/h = 0 m/s
Final velocity = 71 km/h * 0.2778 m/s/km/h = 19.72 m/s
Time is not provided in the problem, so we need to find it. We can use the equation for displacement:
Displacement = (initial velocity * time) + (1/2 * acceleration * time^2)
In this case, the displacement given is 49 m, the initial velocity is 0 m/s, and the final velocity is 19.72 m/s. We need to solve for time.
49 m = (0 m/s * t) + (1/2 * a * t^2)
To solve for time, we need to rearrange the equation:
49 m = 1/2 * a * t^2
Multiplying both sides by 2:
98 m = a * t^2
Now we can find the time:
t^2 = 98 m / a
t = sqrt(98 m / a)
Now we can substitute this value for time back into the equation for change in velocity:
Average acceleration = (19.72 m/s - 0 m/s) / sqrt(98 m / a)
Simplifying the equation:
Average acceleration = 19.72 m/s / sqrt(98 m / a)
To find the displacement at t = 3.0 s, we can use the equation for displacement mentioned earlier:
Displacement = (initial velocity * time) + (1/2 * acceleration * time^2)
In this case, the initial velocity is 0 m/s, the time is 3.0 s, and the acceleration can be derived from the initial equation for average acceleration. Let's call the acceleration we found "a".
So, the acceleration is:
a = 19.72 m/s / sqrt(98 m / a)
Now we can substitute this value for acceleration and the given values into the equation for displacement:
Displacement = (0 m/s * 3.0 s) + (1/2 * a * (3.0 s)^2)
Simplifying the equation:
Displacement = 0 m + 1/2 * a * 9.0 s^2
Displacement = 4.5 a * s^2
Now we substitute the value of "a":
Displacement = 4.5 * (19.72 m/s / sqrt(98 m / a)) * 9.0 s^2
Simplifying the equation and calculating the value of displacement:
Displacement = 80.25 * sqrt(a)
This is the equation for displacement at t = 3.0 s.