A ball of mass 0.3 kg moving with a speed of 6.0 ms-² strikes a wall at an angle of 60° to the wall. It then rebounds at the same speed and the same angle. It is in contact with the wall for 10 ms . Calculate the impulse and the average force.
speed toward wall = 6 sin 60
speed away from wall = 6 sin 60
change in velocity = 12 sin 60
change in momentum = inpulse = .3 * 12 sin 60
force = change of momentum/time = .3 * 12 sin 60 / .010
To calculate the impulse and average force, we can use the equations of motion and the principles of impulse-momentum.
First, let's find the initial and final velocities of the ball after it rebounds from the wall. Since the ball rebounds at the same speed and angle, the initial and final speeds will be the same.
Given:
Mass of the ball, m = 0.3 kg
Initial speed, u = 6.0 m/s
Angle of incidence, θ = 60°
To find the initial and final velocities, we can use trigonometry:
Initial velocity in the x-direction, ux = u * cos(θ)
Initial velocity in the y-direction, uy = u * sin(θ)
Using the given values:
ux = 6.0 m/s * cos(60°) = 6.0 m/s * 0.5 = 3.0 m/s
uy = 6.0 m/s * sin(60°) = 6.0 m/s * √(3/2) = 6.0 m/s * √(3/2) ≈ 5.2 m/s
Since the ball rebounds at the same angle, the final velocity in the y-direction will be the negative of the initial velocity in the y-direction: vy = -uy ≈ -5.2 m/s.
Now, let's find the time of contact, Δt = 10 ms = 0.01 s.
Impulse (J) is defined as the change in momentum, which can be calculated as:
J = m * (Δv), where Δv is the change in velocity.
The change in velocity in the x-direction is given by: Δvx = vx - ux
The change in velocity in the y-direction is given by: Δvy = vy - uy
Using the given values:
Δvx = 0 - 3.0 m/s = -3.0 m/s
Δvy = -5.2 m/s - 5.2 m/s = -10.4 m/s
Therefore:
J = m * (Δv) = m * [(Δvx)i + (Δvy)j] = 0.3 kg * (-3.0 m/s i - 10.4 m/s j) ≈ -3.0 kg·m/s i - 3.12 kg·m/s j
The impulse has both a magnitude and direction, represented by the vector. In this case, the impulse vector points in the opposite direction of the initial velocity, since the ball rebounds.
Next, let's calculate the average force applied to the ball during the collision.
Average force (Favg) is defined as the impulse divided by the time of contact:
Favg = J / Δt
Using the calculated value of J and the given value of Δt:
Favg = (-3.0 kg·m/s i - 3.12 kg·m/s j) / 0.01 s ≈ -300 N i - 312 N j
Therefore, the impulse is approximately -3.0 kg·m/s in the x-direction and -3.12 kg·m/s in the y-direction, and the average force is approximately -300 N in the x-direction and -312 N in the y-direction. The negative sign indicates that the force is in the opposite direction of the initial velocity.