a golf ball of mass 0.045kg is moving in the x-direction with a speed of 9.0 m/s, and a baseball with mass 0.145kg is moving in a negative y-directionwith a speed of 7m/s. what are the magnitude and direction of the total momentum of the system consisting of the two balls?
Using the standard orthogonal unit vectors î and ĵ.
Momentum of the system:
0.045*9.0 î - 0.145*7 ĵ kg∙m/s
Magnitude:
√((0.045*9.0)^2 + (0.145*7)^2) kg∙m/s
Direction: angle measured off x-axis
-arctan((0.145*7)/(0.045*9.0))
To determine the magnitude and direction of the total momentum of the system, we need to find the sum of the momentum vectors of the two balls.
The momentum of an object is given by the equation: momentum (p) = mass (m) * velocity (v).
For the golf ball:
Mass (m1) = 0.045 kg
Velocity (v1) = 9.0 m/s
The momentum of the golf ball can be calculated as:
p1 = m1 * v1 = 0.045 kg * 9.0 m/s
For the baseball:
Mass (m2) = 0.145 kg
Velocity (v2) = -7.0 m/s (negative y-direction)
The momentum of the baseball can be calculated as:
p2 = m2 * v2 = 0.145 kg * (-7.0 m/s)
Now, we can find the total momentum, which is the vector sum of the individual momenta. Since one object is moving in the x-direction and the other in the negative y-direction, we need to add the two vectors using vector addition.
The magnitude of the total momentum can be calculated using the Pythagorean theorem:
Magnitude = √(p1^2 + p2^2)
Direction can be determined using the inverse tangent (arctan) function:
Direction = arctan(p2 / p1)
Let's calculate the magnitude and direction:
p1 = 0.045 kg * 9.0 m/s = 0.405 kg·m/s
p2 = 0.145 kg * (-7.0 m/s) = -1.015 kg·m/s
Magnitude = √(0.405^2 + (-1.015)^2) = √(0.164 + 1.030) = √1.194 = 1.095 kg·m/s
Direction = arctan((-1.015) / 0.405) ≈ -68.67°
Therefore, the magnitude of the total momentum of the system is approximately 1.095 kg·m/s, and the direction is approximately -68.67°.
To find the magnitude and direction of the total momentum of the system consisting of the two balls, we need to find the individual momenta of the golf ball and baseball and then add them together.
The momentum of an object can be calculated by multiplying its mass by its velocity. Mathematically, it can be expressed as:
Momentum = mass * velocity
For the golf ball:
Mass of the golf ball, m1 = 0.045 kg
Velocity of the golf ball, v1 = 9.0 m/s
The momentum of the golf ball, P1 = m1 * v1 = 0.045 kg * 9.0 m/s = 0.405 kg·m/s in the x-direction.
For the baseball:
Mass of the baseball, m2 = 0.145 kg
Velocity of the baseball, v2 = -7 m/s (negative y-direction)
The momentum of the baseball, P2 = m2 * v2 = 0.145 kg * (-7 m/s) = -1.015 kg·m/s in the negative y-direction.
To find the total momentum of the system, we need to add the individual momenta together. Since they are in different directions, we need to consider both magnitude and direction.
Magnitude of the total momentum:
Total momentum = |P1| + |P2|
= |0.405 kg·m/s| + |(-1.015 kg·m/s)|
= 0.405 kg·m/s + 1.015 kg·m/s
= 1.42 kg·m/s
Direction of the total momentum:
Since the golf ball has momentum in the x-direction and the baseball has momentum in the negative y-direction, we have to find the resultant direction using vector addition. We can use the Pythagorean theorem and trigonometry to find the magnitude and direction of the resultant vector.
Magnitude of the resultant vector:
Magnitude of the resultant vector = sqrt((P1)^2 + (P2)^2)
= sqrt((0.405 kg·m/s)^2 + (-1.015 kg·m/s)^2)
= sqrt(0.164025 kg^2·m^2/s^2 + 1.030225 kg^2·m^2/s^2)
= sqrt(1.19425 kg^2·m^2/s^2)
= 1.093 kg·m/s
Direction of the resultant vector:
tan(θ) = (P2)/(P1)
θ = tan^(-1)((P2)/(P1))
θ = tan^(-1)((-1.015 kg·m/s)/(0.405 kg·m/s))
θ = tan^(-1)(-2.506)
θ ≈ -1.191 radians
The magnitude of the total momentum is 1.42 kg·m/s, and the direction is approximately -1.191 radians or in the fourth quadrant when measured counterclockwise from the positive x-axis.