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 89 km/h.
(a) Determine the cheetah's average acceleration during the short sprint.
m/s2
(b) Find its displacement at
t = 3.2 s.
(Assume the cheetah maintains a constant acceleration throughout the sprint.)
m
First, convert km/hr to m/s
(a)v^2 = 2as
(b) s = 1/2 at^2
To determine the cheetah's average acceleration during the short sprint, we can use the equation:
Acceleration = (Final Velocity - Initial Velocity) / Time
First, we need to convert the final velocity from km/h to m/s:
Final Velocity = 89 km/h * (1000 m/1 km) * (1 h/3600 s) = 24.7 m/s
Since the cheetah starts from rest, the initial velocity is 0 m/s.
The time taken for the sprint is not given directly, so we need to calculate it first. We can use the equation of motion:
Displacement = 0.5 * Acceleration * Time^2 + Initial Velocity * Time
Rearranging the equation, we get:
Time = (Final Velocity - Initial Velocity) / Acceleration
Since the cheetah starts from rest, the initial velocity is 0 m/s:
Time = (24.7 m/s - 0 m/s) / Acceleration
Now, we can calculate the time taken for the sprint:
Time = 50 m / (24.7 m/s) = 2.02 s
The average acceleration can be calculated as:
Acceleration = (24.7 m/s - 0 m/s) / 2.02 s ≈ 12.19 m/s²
So, the cheetah's average acceleration during the short sprint is approximately 12.19 m/s².
To find the displacement at t = 3.2 s, we can use the equation of motion:
Displacement = 0.5 * Acceleration * Time^2 + Initial Velocity * Time
Since the cheetah starts from rest, the initial velocity is 0 m/s. Plugging in the values:
Displacement = 0.5 * 12.19 m/s² * (3.2 s)^2 + 0 m/s * 3.2 s
Simplifying the equation:
Displacement = 0.5 * 12.19 m/s² * 10.24 s²
Displacement = 62.72 m
Therefore, the cheetah's displacement at t = 3.2 s is approximately 62.72 m.