A young girl with mass 39.5 kg is sliding on a horizontal, frictionless surface with an initial momentum that is due east and that has magnitude 85.5 kg⋅m/s . Starting at 0 s , a net force with magnitude F=( 8.35 N/s )t and direction due west is applied to the girl. A) At what value of t does the girl have a westward momentum of magnitude 64.5 kg⋅m/s ? B)How much work has been done on the girl by the force in the time interval from t= 0 s to the time calculated in part A? C) What is the magnitude of the acceleration of the girl at the time calculated in part A?
show me what you can do with this.
A) Well, it looks like this girl is sliding and getting a force applied to her. Maybe someone doesn't want her to slide too far, you know, like when you're trying to stop someone from singing karaoke at a party. Anyway, let's find out at what value of t she has a westward momentum of magnitude 64.5 kg⋅m/s.
Since she starts with an initial momentum of magnitude 85.5 kg⋅m/s, and the force applied to her is in the opposite direction (westward), we can set up an equation:
Δp = Ft
Where Δp is the change in momentum and Ft is the impulse applied to the girl. In this case, the impulse is given by the integral of the force over time:
Impulse = ∫(8.35 N/s)t dt
Integrating that will give us the impulse as a function of time. However, we can simplify things by noting that the impulse is the area under a linear graph of force vs. time. Since the force is changing linearly with time (F = (8.35 N/s)t), the graph is a triangle, and the area can be calculated as (1/2) * base * height.
The base of the triangle is the time interval t and the height is the force at that time, which is (8.35 N/s)t. So, the impulse is given by:
Impulse = (1/2) * t * (8.35 N/s)t
Now, we know that the impulse is also equal to the change in momentum, so we can set it equal to the desired westward momentum of magnitude 64.5 kg⋅m/s:
(1/2) * t * (8.35 N/s)t = 64.5 kg⋅m/s
Solving for t is like solving a riddle or a puzzle. You can use trial and error or rely on your math skills. Once you find the value of t, you'll have your answer for part A!
B) Now, let's calculate how much work has been done on the girl by the force in the time interval from t = 0 s to the time calculated in part A.
Work is defined as the product of force and displacement. Since the force is constant (F = (8.35 N/s)t) and the displacement is unknown, we can't directly calculate the work done. However, we can use the work-energy principle to find the work done.
The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. In this case, the initial kinetic energy is 0 since the girl starts from rest, and the final kinetic energy can be calculated using the final velocity.
The final velocity can be found using the equation:
vf = vi + at,
where vi is the initial velocity (0 m/s), a is the acceleration, and t is the time calculated in part A. Once we have the final velocity, we can calculate the final kinetic energy:
Kf = (1/2)mvf^2,
where m is the mass of the girl (39.5 kg).
The work done on the girl is equal to the change in kinetic energy:
Work = Kf - Ki,
where Ki is the initial kinetic energy (0 J).
So, once you find the displacement, you'll be able to calculate the work done!
C) Finally, let's find the magnitude of the acceleration of the girl at the time calculated in part A.
We know that acceleration is the rate of change of velocity, and in this case, velocity is changing with time due to the applied force. We can use Newton's second law of motion to find the acceleration:
F = ma,
where F is the net force and m is the mass of the girl (39.5 kg).
In this case, the net force is given by F = (8.35 N/s)t and a is the acceleration we're trying to find. Rearranging the equation, we have:
a = F/m,
Substituting the values and calculated t, you can determine the magnitude of the acceleration.
I hope that helps, but don't hesitate to ask if you have any more questions or if you need another dose of Clown Bot humor!
A) To find the value of t when the girl has a westward momentum of magnitude 64.5 kg⋅m/s, we can use the principle of conservation of momentum.
We are given that the initial momentum of the girl is 85.5 kg⋅m/s towards the east and the net force applied is in the opposite direction (due west). Since the net force is constant, the change in momentum per unit time will also be constant.
Using the equation:
Net force = Rate of change of momentum
F = Δp/Δt
Where F is the net force, Δp is the change in momentum, and Δt is the change in time.
Initially, the girl's momentum is in the positive x-direction, and after a time t, it becomes 64.5 kg⋅m/s in the negative x-direction (westward).
Therefore, the change in momentum is:
Δp = 64.5 kg⋅m/s - 85.5 kg⋅m/s = -21 kg⋅m/s
Now, we can solve for t:
F = Δp/Δt
8.35 N/s * t = -21 kg⋅m/s
t = (-21 kg⋅m/s) / (8.35 N/s)
t ≈ -2.52 s
Since time cannot be negative, the magnitude of t is 2.52 seconds.
Therefore, at t = 2.52 s, the girl will have a westward momentum of magnitude 64.5 kg⋅m/s.
B) To calculate the work done on the girl by the force in the time interval from t = 0 s to the time calculated in part A (t = 2.52 s), we can use the work-energy principle.
The work done on an object is given by the equation:
Work = Force * Distance
In this case, the force applied is changing with time, so we need to determine the appropriate integral to find the work done.
The work done on the girl is:
Work = ∫F dx
By integrating the force function with respect to the position, we can find the work done.
F = (8.35 N/s) * t (due west)
dx = v dt
Where dx is the change in position (distance) and v is the velocity.
The velocity can be related to momentum as v = p/m.
Substituting in the given values:
F = (8.35 N/s) * t (due west)
dx = (p/m) * dt = (64.5 kg⋅m/s) / (39.5 kg) * dt
Integration limits are from t = 0 s to t = 2.52 s.
Now, we can integrate:
Work = ∫F dx = ∫(8.35 N/s * t)(64.5 kg⋅m/s) / (39.5 kg) dt
Work = ∫(8.35 N/s * t)(1.63 m) dt
Work = 13.6 N⋅m
Therefore, the work done on the girl by the force in the time interval from t = 0 s to t ≈ 2.52 s is approximately 13.6 N⋅m.
C) To find the magnitude of acceleration at the time calculated in part A (t ≈ 2.52 s), we can use the equation:
Net force = mass * acceleration
We are given that the net force is changing with time, and at t = 2.52 s, it is given by:
F = (8.35 N/s) * t
Substituting the given values:
(8.35 N/s) * t = (39.5 kg) * a
a = (8.35 N/s) * t / (39.5 kg)
a ≈ (8.35 N/s) * 2.52 s / (39.5 kg)
a ≈ 0.531 m/s^2
Therefore, the magnitude of acceleration of the girl at the time calculated in part A (t ≈ 2.52 s) is approximately 0.531 m/s^2.
To solve this problem, we will use the laws of motion and the concepts of momentum, force, work, and acceleration.
A) To find the value of t when the girl has a westward momentum of magnitude 64.5 kg⋅m/s, we need to use the principle of conservation of momentum. The initial momentum is due east, so the change in momentum will be from east to west.
According to the conservation of momentum, the initial momentum is equal to the final momentum.
Initial momentum = Final momentum
Mass × Initial velocity = Mass × Final velocity
Given:
Initial momentum (due east) = 85.5 kg⋅m/s
Final momentum (due west) = 64.5 kg⋅m/s
Since mass cancels out, we can write:
Initial velocity = Final velocity
85.5 = 64.5
Solving for t:
F = (8.35 N/s) × t
Force = mass × acceleration
Acceleration = Force / mass
8.35 = (39.5 kg) × (acceleration)
Simplifying:
acceleration = 8.35 / 39.5
Plug in the values to find the acceleration:
acceleration = 0.2113924051 m/s²
To find the value of t, we'll use kinematic equations to calculate the distance traveled by the girl.
Final velocity = Initial velocity + acceleration × t
Since the girl is slowing down, the final velocity will be zero when she reaches the desired momentum.
0 = 64.5 + (0.2113924051 m/s²) × t
Solving for t:
64.5 = 0.2113924051 × t
t = 64.5 / 0.2113924051
The value of t is approximately 305.06 seconds.
B) To find the work done on the girl by the force in the time interval from t = 0 s to the time calculated in part A, we can use the work-energy theorem.
The work done is equal to the change in kinetic energy of the girl.
Work = Change in kinetic energy
Work = Final kinetic energy - Initial kinetic energy
The initial kinetic energy is zero since the initial velocity is zero.
The final kinetic energy can be calculated using the kinematic equation:
Final velocity = Initial velocity + acceleration × t
Final velocity = 0 + (0.2113924051 m/s²) × 305.06 s
The change in kinetic energy is then:
Change in kinetic energy = 1/2 × mass × (Final velocity)² - 1/2 × mass × (Initial velocity)²
Change in kinetic energy = 1/2 × 39.5 kg × [(0 + (0.2113924051 m/s²) × 305.06 s)² - 0²]
Calculate the change in kinetic energy to find the work done on the girl.
C) The magnitude of acceleration of the girl at the time calculated in part A is already calculated as 0.2113924051 m/s².