a gymnast tumbles forward 4.0m, does cartwheels to the left for 6.0m and climbs a vertical rope to a height of 3.0m. What is the magnitude of the gymnast's displacement?
To find the magnitude of the gymnast's displacement, we need to calculate the straight-line distance from the starting point to the ending point, considering both the distance and direction.
First, let's break down the movement into its components:
1. Tumbling forward 4.0m is a displacement in the x-direction (horizontal) with no displacement in the y-direction (vertical).
2. Cartwheeling to the left for 6.0m is a displacement in the y-direction (vertical) with no displacement in the x-direction (horizontal).
3. Climbing a vertical rope to a height of 3.0m is a displacement in the y-direction (vertical) with no displacement in the x-direction (horizontal).
Since the first displacement is in the x-direction and there are no displacements in the y-direction, we can add all the x-components together. Since the other two displacements are in the y-direction, we can add them up separately.
For the x-components:
The displacement in the x-direction is 4.0m.
For the y-components:
The displacement from cartwheeling left is -6.0m (negative since it is in the opposite direction of the positive y-axis).
The displacement from climbing the rope is +3.0m.
Next, we can calculate the overall displacement by finding the vector sum of the x and y-components:
The total x-displacement is 4.0m.
The total y-displacement is -6.0m + 3.0m = -3.0m.
Using the Pythagorean theorem, we can calculate the magnitude of the displacement:
Magnitude = √(x-displacement^2 + y-displacement^2)
Magnitude = √(4.0^2 + (-3.0)^2)
Magnitude = √(16 + 9)
Magnitude = √(25)
Magnitude = 5.0m
Thus, the magnitude of the gymnast's displacement is 5.0m.
To find the magnitude of the gymnast's displacement, we need to find the straight-line distance from the initial position to the final position.
Let's break down the different movements:
1. The gymnast tumbles forward 4.0m. This means their displacement in the forward direction is +4.0m.
2. The gymnast does cartwheels to the left for 6.0m. This means their displacement in the left direction is -6.0m.
3. The gymnast climbs a vertical rope to a height of 3.0m. This means their displacement in the upward direction is +3.0m.
To find the total displacement, we need to add up the displacements along the x-axis (horizontal direction) and the y-axis (vertical direction).
Total displacement along the x-axis = +4.0m - 6.0m = -2.0m (since the gymnast moved left)
Total displacement along the y-axis = +3.0m
To get the magnitude of the displacement, we use the Pythagorean theorem: magnitude = √(x^2 + y^2)
Substituting the values:
magnitude = √((-2.0m)^2 + (3.0m)^2)
magnitude = √(4.0m^2 + 9.0m^2)
magnitude = √(13.0m^2)
magnitude = 3.61m
Therefore, the magnitude of the gymnast's displacement is approximately 3.61m.