A particle (mass = 5.0 g, charge = 40 mC) moves in a region of space where the electric
field is uniform and is given by Ex = 2.5 N/C, Ey = Ez = 0. If the velocity of the particle at
t = 0 is given by vy = 50 m/s, vx = vz = 0, what is the speed of the particle at t = 2.0 s?
A particle of mass 0.000137 g and charge21 mC moves in a region of space where the electric field is uniform and is 4.3 N/C in the x direction and zero in the y and z direction.If the initial velocity of the particle is given by vy= 1.3×105m/s,vx=vz= 0, what is the speed of the particle at 0.5 s?Answer in units of m/s.
To find the speed of the particle at t = 2.0 s, we can use the equations of motion for a charged particle in an electric field.
Step 1: Calculate the force on the particle
The force on a charged particle in an electric field is given by F = q * E, where F is the force, q is the charge, and E is the electric field. In this case, the force on the particle is given by F = (40 * 10^-3 C) * (2.5 N/C). Calculate the force to get the value.
Step 2: Calculate the acceleration of the particle
The acceleration of the particle is given by the equation F = m * a, where F is the force, m is the mass, and a is the acceleration. Rearranging the equation, we can solve for a. Substitute the values of force and mass to calculate the acceleration.
Step 3: Calculate the change in velocity
At time t = 2.0 s, the particle has been accelerating for 2.0 seconds. The change in velocity is given by the equation Δv = a * t. Substitute the value of acceleration and time to calculate the change in velocity.
Step 4: Calculate the final velocity
The final velocity of the particle can be obtained by adding the change in velocity to the initial velocity. In this case, the initial velocity vy = 50 m/s, and the change in velocity is calculated in the previous step. Add these values to get the final velocity.
Step 5: Calculate the speed
The speed of the particle is the magnitude of the final velocity. Since the electric field is only in the x-direction and the velocity in the y-direction, the speed can be calculated as √(vx^2 + vy^2 + vz^2). Calculate the speed value to get the final answer.
Note: Given that the electric field is uniform and only in the x-direction, the y and z components of the velocity will remain constant over time.
To find the speed of the particle at t = 2.0 s, we need to calculate the net force acting on the particle at that time and then use it to determine the acceleration. Once we know the acceleration, we can find the change in velocity and, consequently, the speed of the particle.
First, let's calculate the electric force on the particle using the formula F = q * E, where F is the force, q is the charge, and E is the electric field. In this case, the electric field is given as Ex = 2.5 N/C, Ey = Ez = 0, and the charge is 40 mC (\(1 \, \text{C} = 10^3 \, \text{mC}\)), so the electric force in the x-direction (Fex) can be calculated as follows:
Fex = q * Ex
= (40 * 10^(-3) C) * (2.5 N/C)
= 0.1 N
Since the electric force is the only force acting on the particle in this scenario, the net force on the particle (Fnet) is equal to the electric force in the x-direction (Fex).
Next, we can calculate the acceleration of the particle using Newton's second law (F = m * a), where F is the net force and m is the mass of the particle. The mass of the particle is given as 5.0 g (\(1 \, \text{g} = 10^{-3} \, \text{kg}\)), so we get:
Fnet = m * a
0.1 N = (5.0 * 10^(-3) kg) * a
Solving for a, we find a = 0.1 N / (5.0 * 10^(-3) kg) = 20 m/s^2.
Now that we know the acceleration, we can determine the change in velocity of the particle using the kinematic equation:
vf = vi + a * t,
where vf is the final velocity, vi is the initial velocity, a is the acceleration, and t is the time interval. In this case, the initial velocity vy = 50 m/s and vx = vz = 0 m/s, so we can consider only the velocity in the y-direction.
vyf = vyi + a * t
vyf = 50 m/s + (20 m/s^2) * (2.0 s)
vyf = 50 m/s + 40 m/s
vyf = 90 m/s
Finally, we can calculate the speed of the particle by finding the magnitude of the final velocity:
speed = sqrt((vx^2 + vy^2 + vz^2))
speed = sqrt((0^2 + 90^2 + 0^2))
speed = sqrt(0 + 8100 + 0)
speed = sqrt(8100)
speed = 90 m/s
Therefore, the speed of the particle at t = 2.0 s is 90 m/s.