A 10 g particle is moving to the left at 20 m/s. How much work must be done on the particle to cause it to move to the right at 46 m/s?
To find the work done on the particle, we need to understand the concept of work and the work-energy principle.
Work is defined as the transfer of energy that occurs when a force acts on an object and causes it to move in the direction of the force. Mathematically, work is given by the equation:
W = F * d * cosθ
Where W is the work done, F is the applied force, d is the displacement, and θ is the angle between the force and displacement vectors.
In this case, the particle initially moves to the left at 20 m/s, and we want it to move to the right at 46 m/s. Since the direction of motion changes, it implies that an external force must be applied to reverse the particle's velocity.
To calculate the work done on the particle, we need to determine the change in kinetic energy. The work-energy principle states that the work done on an object is equal to the change in its kinetic energy:
W = ΔKE
The change in kinetic energy is given by:
ΔKE = KE_final - KE_initial
Let's calculate the initial and final kinetic energies:
KE_initial = 1/2 * m * (v_initial)^2
= 1/2 * 10 g * (20 m/s)^2
KE_final = 1/2 * m * (v_final)^2
= 1/2 * 10 g * (46 m/s)^2
Remember to convert the mass from grams to kilograms (1 kg = 1000 g).
Now we can find the change in kinetic energy:
ΔKE = KE_final - KE_initial
Substitute the values and solve for ΔKE. The value of ΔKE will give us the work done on the particle to change its velocity.
I'll let you perform the calculations to find the value of ΔKE and the work done on the particle.
To calculate the work done on the particle, we can use the work-energy principle which states that the work done on an object is equal to the change in its kinetic energy.
The initial kinetic energy of the particle is given by:
KE_initial = 1/2 * m * v_initial^2
where m is the mass of the particle (10 g = 0.01 kg) and v_initial is the initial velocity (-20 m/s).
Substituting the given values:
KE_initial = 1/2 * 0.01 kg * (-20 m/s)^2
= 0.01 kg * 400 m^2/s^2
= 4 J (joules)
The final kinetic energy of the particle is given by:
KE_final = 1/2 * m * v_final^2
where v_final is the final velocity (46 m/s).
Substituting the given values:
KE_final = 1/2 * 0.01 kg * (46 m/s)^2
= 0.01 kg * 2116 m^2/s^2
= 21.16 J
The change in kinetic energy is given by:
change in KE = KE_final - KE_initial
= 21.16 J - 4 J
= 17.16 J
Therefore, the amount of work done on the particle to cause it to move to the right at 46 m/s is 17.16 joules.