An ant walks on a piece of graph paper straight along the x axis a distance of 10.0 cm in 2.00 s. It then turns left 30.0 degrees and walks in a straight line another 10.0 cm in 1.90 s. Finally, it turns another 70.0 degrees to the left and walks another 10.0 cm in 1.40 s.
A. Determine the x component of the ant's average velocity.
B. Determine the y component of the ant's average velocity.
C. Determine the magnitude of ant's average velocity.
D. Determine the direction of ant's average velocity.
A. Divide the total change in the x coordinate by (2.0+1.9+1.4 =) 5.3 s
B. Divide the total change in the y coordinate by 5.3 s.
C. Use the Pythagorean Theorem with the previous two results
D. arctan (B answer)/(A answer)
counterclockwise from +x
To solve this problem, we can break down the ant's motion into x and y components and then calculate the average velocity in each component separately. The x component represents motion along the x-axis, and the y component represents motion along the y-axis.
Let's start by calculating the x and y components.
A. Determining the x component of the ant's average velocity:
The ant walks straight along the x-axis for 10.0 cm in 2.00 s. The x-component of velocity is simply the displacement along the x-axis divided by the time.
x = displacement along x-axis / time
x = 10.0 cm / 2.00 s
x = 5.0 cm/s
Therefore, the x component of the ant's average velocity is 5.0 cm/s.
B. Determining the y component of the ant's average velocity:
The ant turns left 30.0 degrees and walks for 10.0 cm in 1.90 s. To calculate the y component, we need to find the displacement along the y-axis.
The displacement along the y-axis is given by the formula:
displacement along y-axis = distance * sin(angle)
Displacement along y-axis = 10.0 cm * sin(30.0 degrees)
Displacement along y-axis = 10.0 cm * 0.5
Displacement along y-axis = 5.0 cm
Now we can calculate the y component of the velocity:
y = displacement along y-axis / time
y = 5.0 cm / 1.90 s
y = 2.63 cm/s
Therefore, the y component of the ant's average velocity is 2.63 cm/s.
C. Determining the magnitude of the ant's average velocity:
The magnitude of the average velocity is given by the formula:
magnitude = sqrt( x^2 + y^2)
magnitude = sqrt( (5.0 cm/s)^2 + (2.63 cm/s)^2 )
magnitude = sqrt( 25.0 cm^2/s^2 + 6.91 cm^2/s^2 )
magnitude = sqrt( 31.91 cm^2/s^2 )
magnitude ≈ 5.65 cm/s
Therefore, the magnitude of the ant's average velocity is approximately 5.65 cm/s.
D. Determining the direction of the ant's average velocity:
The direction of the ant's average velocity can be found using trigonometry. We can use the arctan function to find the angle.
direction = arctan( y / x )
direction = arctan( 2.63 cm/s / 5.0 cm/s )
direction ≈ 28.07 degrees
Therefore, the direction of the ant's average velocity is approximately 28.07 degrees.