Mike throws a ball upward & toward the east at a 20.0 angle with a speed of 51.0 . Nancy drives east past Mike at 15.0 at the instant he releases the ball.
What is the ball's initial angle in Nancy's reference frame?
v(x) = v•cosα = 51•cos20 = 47.9 m/s.
Relative velocity
v(x-rel)= 47.9-15 = 32.9 m/s.
v(y) = v•sin20 =17.4 m/s.
tanφ = v(y)/v(x-rel) = 17.4/32.9 = 0.53.
φ = 27.9º
To find the ball's initial angle in Nancy's reference frame, we need to consider their relative motion.
First, let's assume that Nancy is at rest. In this frame of reference, Mike's velocity will appear to be the sum of his actual velocity and Nancy's velocity because both of them are moving in the same direction (east).
Mike's actual velocity (u) = 51.0 m/s (thrown upward & toward the east)
Nancy's velocity (v) = 15.0 m/s (east)
Therefore, the relative velocity of Mike with respect to Nancy is:
Relative velocity (vrel) = u + v
vrel = 51.0 m/s + 15.0 m/s
vrel = 66.0 m/s (east)
Now, we can use the relative velocity to find the initial angle of the ball in Nancy's reference frame.
The initial angle (θ') of the ball in Nancy's reference frame can be found using the tangent function, which is defined as:
tan(θ') = (vertical component of relative velocity) / (horizontal component of relative velocity)
Since the ball is thrown upward & toward the east, the vertical component of the relative velocity is positive, and the horizontal component is also positive.
Vertical component of relative velocity = vrel * sin(θ)
Horizontal component of relative velocity = vrel * cos(θ)
The ratio of these two components gives the tangent of the angle θ' in Nancy's reference frame.
tan(θ') = (vrel * sin(θ)) / (vrel * cos(θ))
tan(θ') = sin(θ) / cos(θ)
tan(θ') = tan(θ)
Since the tangent function is the same for both angles, the initial angle of the ball in Nancy's reference frame (θ') is equal to the initial angle of the ball relative to the ground (θ).
Therefore, the ball's initial angle in Nancy's reference frame is 20.0 degrees.
To find the ball's initial angle in Nancy's reference frame, we need to consider the relative motion between Mike and Nancy.
First, let's break down the motion of the ball and Nancy individually:
1. Ball's motion:
- The ball is thrown upward and toward the east with an angle of 20.0° and a speed of 51.0 m/s.
2. Nancy's motion:
- Nancy drives east at a speed of 15.0 m/s.
Now, let's determine the relative motion of the ball with respect to Nancy:
- Since Nancy is moving eastward, she will see the ball's initial motion slightly differently than in Mike's reference frame.
To determine the ball's initial angle in Nancy's reference frame, we need to subtract Nancy's velocity from the ball's initial velocity component in the east direction.
Given:
- Ball's initial angle = 20.0°
- Ball's initial velocity = 51.0 m/s
- Nancy's velocity = 15.0 m/s
Now, let's perform the calculations:
1. Break down the ball's initial velocity into its components:
- Velocity in the east direction (Vx) = 51.0 m/s * cos(20.0°)
- Velocity in the north direction (Vy) = 51.0 m/s * sin(20.0°)
2. Subtract Nancy's velocity component in the east direction from the ball's eastward velocity component:
- Relative velocity in the east direction = Vx - Nancy's velocity
Finally, use the relative velocity components to find the ball's initial angle in Nancy's reference frame:
- Ball's initial angle in Nancy's reference frame = arctan(Vy / (Relative velocity in the east direction))
Now, substitute the given values into the equations and calculate the results.