In getting ready to slam-dunk the ball, a basketball player starts from rest and sprints to a speed of 5.17 m/s in 1.29 s. Assuming that the player accelerates uniformly, determine the distance he runs.
v=v₀+at
v₀=0
v=at
a=v/t
s=at²/2
To find the distance the basketball player runs, we need to use the equation for distance:
distance = initial velocity * time + (1/2) * acceleration * time^2
Since the player starts from rest, the initial velocity (u) is 0 m/s. The time (t) is given as 1.29 s. We need to find the acceleration (a) first.
We can use the equation:
final velocity = initial velocity + (acceleration * time)
The final velocity (v) is given as 5.17 m/s. Rearranging the equation, we can solve for acceleration:
acceleration = (final velocity - initial velocity) / time
acceleration = (5.17 m/s - 0 m/s) / 1.29 s
acceleration = 5.17 m/s / 1.29 s
acceleration = 4 m/s^2 (approx)
Substituting the values into the distance equation:
distance = 0 m/s * 1.29 s + (1/2) * 4 m/s^2 * (1.29 s)^2
distance = 0 m + (1/2) * 4 m/s^2 * 1.6641 s^2
distance = 0 m + 3.3282 m
distance = 3.3282 m
Therefore, the basketball player runs a distance of approximately 3.33 meters.
To determine the distance the basketball player runs, we can use the equation of motion known as the "displacement equation":
s = ut + (1/2)at^2
Where:
s = displacement (distance)
u = initial velocity (rest or 0 m/s in this case)
t = time taken to reach the final velocity (1.29 s in this case)
a = acceleration (to be determined)
Since the player starts from rest (u = 0 m/s), the equation simplifies to:
s = (1/2)at^2
To find the acceleration (a), we can use the formula:
a = (v - u) / t
Where:
v = final velocity (5.17 m/s)
u = initial velocity (0 m/s)
t = time taken (1.29 s)
Substituting the given values into the equation:
a = (5.17 - 0) / 1.29
a = 5.17 / 1.29
a ≈ 4 m/s²
Now, we can substitute the calculated acceleration into the displacement equation:
s = (1/2)at^2
s = (1/2) * 4 * (1.29)^2
s = 2 * 4 * 1.66
s ≈ 13.28 m
Therefore, the basketball player runs approximately 13.28 meters.