To find the centripetal acceleration, tangential acceleration, and resultant acceleration, we need to understand their definitions and formulas.
1. Centripetal Acceleration:
Centripetal acceleration is the acceleration experienced by an object moving in a circular path. It constantly changes the direction of the object towards the center of the circle. The formula for centripetal acceleration is given by:
a_c = (v^2) / r,
where a_c is the centripetal acceleration, v is the velocity, and r is the radius of the circular path.
2. Tangential Acceleration:
Tangential acceleration is the acceleration in the direction tangent to the circular path. It is caused by changes in speed. The formula for tangential acceleration is:
a_t = dv / dt,
where a_t is the tangential acceleration, dv is the change in velocity, and dt is the change in time.
3. Resultant Acceleration:
Resultant acceleration is the overall acceleration experienced by the object, considering both the centripetal and tangential accelerations. It can be calculated using the Pythagorean theorem:
a_r = sqrt((a_c)^2 + (a_t)^2),
where a_r is the resultant acceleration.
Now, let's apply these concepts to solve the problem.
Given:
Length of the vine (radius), r = 4m
Angle with the vertical, θ = 20 degrees
Speed, v = 5m/s
1. Centripetal Acceleration:
To find the centripetal acceleration, we need to calculate the radius of the circular path. Since Tarzan swings from a vine, the length of the vine (4m) is the radius of the circle. Therefore, r = 4m.
Using the formula: a_c = (v^2) / r,
Substituting the given values, we get: a_c = (5^2) / 4 m/s^2
Simplifying, we find: a_c = 25 / 4 m/s^2
2. Tangential Acceleration:
In this case, there is no information provided on the change in velocity or time. So, we assume that the tangential acceleration is zero.
3. Resultant Acceleration:
Using the formula: a_r = sqrt((a_c)^2 + (a_t)^2),
Since a_t = 0 (no tangential acceleration), the equation becomes: a_r = sqrt((a_c)^2 + 0)
Simplyfying, we find: a_r = sqrt((a_c)^2)
Thus, a_r = |a_c| (taking the positive value because acceleration is always positive).
Therefore, the centripetal acceleration is 25/4 m/s^2, the tangential acceleration is 0 m/s^2, and the resultant acceleration is 25/4 m/s^2.