body of mass 2kg falls freely from rest through a height of 50m and comes to rest having penetrated 5.0cm of sand.
Calculate.
(I) The velocity with which the ball hits the sand.
(II) The time taken in falling.
(III) The average force exerted by the sand in bringing the body to rest.
1
1. V^2 = Vo^2 + 2g*h = 0 + 19.6*50 =
V = ?
2. 0.5g*t^2 = 50.
4.9t^2 = 50,
3. F = M*g = 2 * 9.8 =
3. No, the average force is the change in momentum divided by the time to stop m V / tstop
where tstop is the stopping time
(1/2) V tstop = 0.05 meters
Given:
Mass of body (m) = 2 kg
Height (h) = 50 m
Penetration (d) = 5.0 cm = 0.05 m
(I) To calculate the velocity with which the ball hits the sand, we can use the conservation of mechanical energy. The initial potential energy (mgh) is converted into kinetic energy (1/2 mv^2) when the ball hits the sand, neglecting air resistance.
Using the equation:
Potential Energy = Kinetic Energy
mgh = 1/2 mv^2
Rearranging the equation, we get:
v^2 = 2gh
Substituting the given values:
v^2 = 2 * 9.8 m/s^2 * 50 m
Calculating:
v^2 = 980 m^2/s^2
v ≈ 31.3 m/s
Therefore, the velocity with which the ball hits the sand is approximately 31.3 m/s.
(II) The time taken in falling can be calculated using the formula for free fall:
h = 1/2 * g * t^2
Rearranging the equation, we get:
t = √(2h/g)
Substituting the given values:
t = √(2 * 50 m / 9.8 m/s^2)
Calculating:
t ≈ √(10.2) ≈ 3.2 s
Therefore, the time taken in falling is approximately 3.2 seconds.
(III) The average force exerted by the sand in bringing the body to rest can be calculated using the formula:
Force = (mass * final velocity - initial velocity * mass) / time
Since the body comes to rest, the final velocity is 0 m/s.
Substituting the given values:
Force = (2 kg * 0 m/s - 31.3 m/s * 2 kg) / 3.2 s
Calculating:
Force ≈ -98.1 N
Since the force is negative, it indicates that the direction of the force exerted by the sand is opposite to the direction of motion.
Therefore, the average force exerted by the sand in bringing the body to rest is approximately 98.1 N in the opposite direction.
To solve this problem, we can use the principles of conservation of energy and motion equations. Let's break down the problem step by step:
(I) The velocity with which the ball hits the sand:
1. First, let's calculate the initial potential energy of the ball when it is at a height of 50m. The potential energy is given by the formula: potential energy = mass x gravity x height
Potential energy = 2kg x 9.8m/s^2 x 50m = 980 Joules
2. At the lowest point, all the potential energy is converted into kinetic energy. Therefore, the kinetic energy can be calculated using the formula: kinetic energy = 0.5 x mass x velocity^2
Set the potential energy equal to the kinetic energy and solve for velocity:
980 J = 0.5 x 2kg x velocity^2
velocity^2 = (980 J) / (1 kg)
velocity^2 = 980 m^2/s^2
velocity = sqrt(980) m/s
velocity ≈ 31.3 m/s
Therefore, the velocity with which the ball hits the sand is approximately 31.3 m/s.
(II) The time taken in falling:
We can calculate the time taken using the motion equation: distance = initial velocity x time + 0.5 x acceleration x time^2
In this case, the body is in free fall, so the initial velocity is 0 m/s, and the distance is 50m.
50m = 0.5 x 9.8m/s^2 x time^2
time^2 = (50m) / (0.5 x 9.8m/s^2)
time^2 = 10.2s^2
time = sqrt(10.2) s
time ≈ 3.19 seconds
Therefore, the time taken for the body to fall is approximately 3.19 seconds.
(III) The average force exerted by the sand in bringing the body to rest:
To find the average force exerted by the sand, we need to calculate the change in momentum of the object. The formula for momentum is: momentum = mass x velocity
The initial momentum of the object is zero since it starts from rest. The final momentum can be calculated using the formula: final momentum = mass x final velocity
The change in momentum is given by: change in momentum = final momentum - initial momentum.
Since the final momentum is zero, the change in momentum is simply the momentum at the start.
Change in momentum = mass x velocity = 2kg x 31.3 m/s = 62.6 kg*m/s
To calculate the average force, we can use the formula: average force = change in momentum / time
average force = (62.6 kg*m/s) / (3.19 s)
average force ≈ 19.6 N
Therefore, the average force exerted by the sand in bringing the body to rest is approximately 19.6 Newtons.