An 85 kg jogger is heading due east at a speed of 1.7 m/s. A 55 kg jogger is heading 40° north of east at a speed of 3.4 m/s. Find the magnitude and direction of the sum of momenta of the two joggers.
1. An 85-kg jogger is heading due east at a speed of 2.0 m/s. A 55-kg jogger is heading 32 north of east at a speed of 3.0 m/s. Find the magnitude and direction of the sum of the momenta of the two joggers.
Well, it seems like these joggers are heading towards a collision course! Brace yourself!
To find the magnitude of the sum of their momenta, we'll need to calculate their individual momenta first. The momentum of an object is given by the formula: momentum = mass × velocity.
For the first jogger:
Momenta1 = mass1 × velocity1 = 85 kg × 1.7 m/s = 144.5 kg·m/s (approximately)
For the second jogger:
Momenta2 = mass2 × velocity2 = 55 kg × 3.4 m/s = 187 kg·m/s (approximately)
Now, to find the magnitude of the sum of their momenta, we can simply add them up:
Magnitude of the sum of momenta = Momenta1 + Momenta2 = 144.5 kg·m/s + 187 kg·m/s = 331.5 kg·m/s (approximately)
That's quite a hefty momentum they've got going on!
Now, let's find the direction. Since one jogger is heading due east, and the other jogger is heading 40° north of east, we need to combine these two directions.
If we take east as 0° and north as 90°, then the second jogger is heading at an angle of 40° + 90° = 130°.
So, the direction of the sum of their momenta is 130° north of east.
And there you have it! The magnitude of the sum of their momenta is approximately 331.5 kg·m/s, and the direction is 130° north of east. Stay safe and keep those joggers away from me!
To find the magnitude and direction of the sum of momenta of the two joggers, we can first calculate each jogger's momentum vectors and then add them together.
The momentum of an object can be calculated using the formula:
Momentum = mass * velocity
For the first jogger:
Mass of the first jogger, m1 = 85 kg
Velocity of the first jogger, v1 = 1.7 m/s
Momentum of the first jogger, p1 = m1 * v1
For the second jogger:
Mass of the second jogger, m2 = 55 kg
Velocity of the second jogger, v2 = 3.4 m/s
Momentum of the second jogger, p2 = m2 * v2
Now, let's calculate the momentum vectors:
For the first jogger:
p1 = (85 kg) * (1.7 m/s) = 144.5 kg·m/s
For the second jogger:
p2 = (55 kg) * (3.4 m/s) = 187 kg·m/s
To find the magnitude and direction of the sum of the momenta, we can add the two momentum vectors together:
Magnitude of the sum of the momenta, |p_sum| = |p1 + p2|
Direction of the sum of the momenta can be calculated using trigonometry.
We know that the second jogger is heading 40° north of east. Therefore, the angle between the vectors is 40°.
Using vector addition, the sum of the momenta can be calculated as follows:
p_sum = √(p1^2 + p2^2 + 2 * p1 * p2 * cos(θ))
Where θ is the angle between the vectors (40°).
p_sum = √((144.5 kg·m/s)^2 + (187 kg·m/s)^2 + 2 * (144.5 kg·m/s) * (187 kg·m/s) * cos(40°))
Calculating the above equation will give the magnitude of the sum of the momenta.
To find the direction, we can use the inverse tangent function:
Direction = tan^(-1)((p2 * sin(θ)) / (p1 + p2 * cos(θ)))
where θ is the angle between the vectors (40°).
Calculating the above equation will give the direction of the sum of the momenta.
Please note that the final result will be in appropriate SI units.
To find the magnitude and direction of the sum of momenta of the two joggers, we first need to calculate the individual momenta for each jogger:
The momentum (p) of an object is given by the formula:
p = m * v
Where:
p = momentum
m = mass
v = velocity
For the first jogger with a mass of 85 kg and a velocity of 1.7 m/s, we can calculate the momentum as follows:
p1 = 85 kg * 1.7 m/s = 144.5 kg·m/s (East)
For the second jogger with a mass of 55 kg and a velocity of 3.4 m/s, we need to consider the angle of 40 degrees north of east. To calculate the horizontal component of the velocity, we need to find the cosine of the angle and multiply it by the magnitude of the velocity:
v2_horizontal = 3.4 m/s * cos(40°) ≈ 2.58 m/s
The horizontal component of the velocity is heading due east. The vertical component is heading north, but we only need the horizontal component to calculate the momentum.
Now we can calculate the momentum of the second jogger:
p2 = 55 kg * 2.58 m/s ≈ 141.9 kg·m/s (East)
To find the magnitude and direction of the sum of the momenta, we can add the horizontal components and vertical components separately.
Horizontal Component:
p_horizontal = p1_horizontal + p2_horizontal
p_horizontal = 144.5 kg·m/s + 141.9 kg·m/s ≈ 286.4 kg·m/s (East)
Vertical Component:
p_vertical = p2_vertical
p_vertical = 0 kg·m/s (North)
The magnitude of the sum of the momenta can be calculated using the Pythagorean theorem:
|p| = sqrt(p_horizontal^2 + p_vertical^2)
|p| = sqrt((286.4 kg·m/s)^2 + (0 kg·m/s)^2)
|p| = sqrt(82138.56 kg^2·m^2/s^2) ≈ 286.52 kg·m/s
The direction of the sum of the momenta can be found by taking the inverse tangent of the vertical component divided by the horizontal component:
θ = atan(p_vertical / p_horizontal)
θ = atan(0 kg·m/s / 286.4 kg·m/s) = atan(0) = 0°
Therefore, the magnitude of the sum of the momenta is approximately 286.52 kg·m/s, and the direction is due east.
The x component of the resultant/sun (combined) momentum is
85*1.7 + 55*3.4 cos 40)= Px
Compute it.
The y component of the resultant/sum is
55*3.4 sin 40 = Py
Compute that.
The magnitude is sqrt(Px^2 + Py^2)
The angle (north from east) is
tan^-1 Py/Px
Do the numbers