A ball of mass 0.1kg dropped from a building of 8m high unto a hard floor bounces back to a height of 2m. Calculate the change in momentum , if the ball is in constant with the floor of 0.1s, what is the force exerted on the body . (g = 10ms)
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To calculate the change in momentum, we can use the formula:
Change in momentum = final momentum - initial momentum
The momentum of an object is given by the formula:
Momentum = mass x velocity
Given that the mass of the ball is 0.1 kg, and the acceleration due to gravity is 10 m/s^2:
1. Calculate the initial momentum before the ball hits the floor:
Initial velocity = sqrt(2gh), where g is the acceleration due to gravity and h is the height of the building.
Initial velocity = sqrt(2 x 10 x 8) = sqrt(160) = 12.65 m/s
Initial momentum = mass x initial velocity = 0.1 kg x 12.65 m/s = 1.265 kg·m/s
2. Calculate the final momentum after the ball bounces back to a height of 2m:
Final velocity = sqrt(2gh), where g is the acceleration due to gravity and h is the height the ball bounces back to.
Final velocity = sqrt(2 x 10 x 2) = sqrt(40) = 6.32 m/s
Final momentum = mass x final velocity = 0.1 kg x 6.32 m/s = 0.632 kg·m/s
3. Calculate the change in momentum:
Change in momentum = final momentum - initial momentum
Change in momentum = 0.632 kg·m/s - 1.265 kg·m/s = -0.633 kg·m/s
The negative sign indicates that the direction of momentum has reversed.
Now, let's calculate the force exerted on the ball. We can use the formula:
Force = (Change in momentum) / (time)
Given that the time of contact with the floor is 0.1 s:
Force = (-0.633 kg·m/s) / (0.1 s)
Force = -6.33 N
The negative sign indicates that the force is in the opposite direction of the initial velocity.
To calculate the change in momentum, we need to determine the initial and final momentums of the ball.
1. Calculating the initial momentum:
Initial momentum (before bounce) = mass * initial velocity
Since the ball is dropped from rest, the initial velocity is 0 m/s.
So, initial momentum = 0.1 kg * 0 m/s = 0 kg*m/s
2. Calculating the final momentum:
Final momentum (after bounce) = mass * final velocity
To find the final velocity, we can use the concept of conservation of mechanical energy. Since the ball is in free fall initially and reaches a height of 2m after bouncing, the potential energy at the top should be equal to the kinetic energy at the bottom.
Potential energy at the top = m * g * height
Kinetic energy at the bottom = 0.5 * m * velocity^2
Since the potential energy at the top is converted into kinetic energy at the bottom, we can equate the two:
m * g * height = 0.5 * m * velocity^2
Substituting the given values:
0.1 kg * 10 m/s^2 * 2 m = 0.5 * 0.1 kg * velocity^2
Simplifying the equation:
20 = 0.05 * velocity^2
velocity^2 = 20 / 0.05 = 400
velocity = sqrt(400) = 20 m/s
Final momentum = 0.1 kg * 20 m/s = 2 kg*m/s
3. Calculating the change in momentum:
Change in momentum = Final momentum - Initial momentum
Change in momentum = 2 kg*m/s - 0 kg*m/s
Change in momentum = 2 kg*m/s
Now, let's move on to calculating the force exerted on the body during the contact of 0.1s.
Force can be calculated using the formula:
Force = Change in momentum / Time
Substituting the given values:
Force = 2 kg*m/s / 0.1 s
Force = 20 N
Therefore, the change in momentum is 2 kg*m/s, and the force exerted on the body is 20 Newtons.
a. V1^2 = Vo^2 + 2g*h.
V1^2 = 0 + 20*8, V1 = 12.65 m/s.
V^2 = V2^2 + 2g*h.
0 = V2^2 - 20*2, V2 = 6.32 m/s.
M*V1-M*V2 = 0.1*12.65 - 0.1*6.32 = 0.633 = Change in momentum.
b. V = V2 + a*t.
0 = 6.32 - a*0.1, a = 63.2 m/s^2.
F = M*a = 0.1 * 63.2 = 6.32 N.