Question

A 500 g particle has velocity vx = -9.0 m/s at t = -2 s. Force

Fx=(4−t2)N, where t is in s, is exerted is exerted on the particle
between t=−2s and t = 2 s. This force increases from 0 N at t = -2

Answer 1

The change in momentum of the particle from t = -2 s to t = 2 s due to the varying force
`\( F_x = (4 - t^2) \, \text{N} \)` is 48 kg m/s.

Step-by-step explanation:

Initially, at t = -2 s, the particle has a velocity of
`\( v_x = -9.0 \, \text{m/s} \)` and a mass of 500 g, which is
`\( 0.5 \, \text{kg} \)`. The force acting on the particle can be integrated over the time interval from t = -2 s to t = 2 s to find the change in momentum using the equation:

`\[ \Delta p = \int_(t_i)^(t_f) F_x \, dt \]`

Integrating the force equation
`\( F_x = (4 - t^2) \)N` with respect to time over the given interval:

`\[ \Delta p = \int_(-2)^(2) (4 - t^2) \, dt \]`

`\[ = \left[ 4t - (t^3)/(3) \right]_(-2)^(2) \]`

`\[ = \left( 8 - (8)/(3) \right) - \left( -8 + (8)/(3) \right) \]`

Therefore, the change in momentum of the particle from t = -2 s to t = 2 s due to the varying force is
`\( (64)/(3) \)` kg m/s or approximately 21.33 kg m/s.

This calculation illustrates the change in momentum caused by the force
`\( F_x = (4 - t^2) \)N` over the given time interval. The integration of the force equation with respect to time helps determ
`\[ = (32)/(3) + (32)/(3) = (64)/(3) \, \text{kg m/s} \]`ine the net change in momentum, which indicates the effect of the varying force on the particle's motion. The resulting value, 64/3 kg m/s, signifies the total change in momentum experienced by the 500 g particle over the specified time period.

Answer 2

The momentum of a particle can be calculated by multiplying its mass by its velocity. In this case, the momentum of the particle is 10.0i + 20.0tj kg.m/s. The net force acting on the particle can be found by using Newton's second law of motion, which states that force equals mass times acceleration.

Step-by-step explanation:

The momentum of a particle is given by the formula:

p = m × v

Where:
p = momentum
m = mass of the particle
v = velocity of the particle

In this case, the mass of the particle is 5.0 kg and the velocity as a function of time is given by v(t) = 2.0i + 4.0tj m/s.

To find the momentum as a function of time, we substitute the values of mass and velocity into the formula:

p(t) = (5.0 kg) × (2.0i + 4.0tj m/s)

Simplifying the expression, we get:

p(t) = 10.0i + 20.0tj kg.m/s

The net force acting on the particle can be found using Newton's second law of motion:

F = ma

Where:
F = force
m = mass of the particle
a = acceleration of the particle

In this case, the mass of the particle is 5.0 kg and the acceleration is given by the derivative of the velocity function:

a(t) = ∂v/∂t = 4.0j m/s²

Therefore, the net force acting on the particle is:

F = (5.0 kg)(4.0j m/s²)