A 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.
students
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 =
V^2=v0^2+2g*h=0+19.6*50=v=?
To calculate the answers, we can use the principles of motion and energy conservation.
(I) The velocity with which the ball hits the sand:
Using the equation for gravitational potential energy:
Potential Energy (PE) = mass * gravity * height
PE = 2kg * 9.8 m/s^2 * 50m = 980 Joules
Since the ball comes to rest after penetrating 5.0cm of sand, all the initial potential energy is converted into kinetic energy and the work done by the sand against the ball's motion:
Kinetic Energy (KE) = 1/2 * mass * velocity^2
KE = 1/2 * 2kg * velocity^2
Since the initial potential energy equals the final kinetic energy:
980 J = 1/2 * 2kg * velocity^2
980 J = velocity^2
Taking the square root of both sides:
velocity = √980 m/s = 31.3 m/s
(II) The time taken in falling:
We can use the equation for free fall:
height = (1/2) * gravity * time^2
Plugging in the values:
50m = (1/2) * 9.8 m/s^2 * time^2
Simplifying:
time^2 = 50m / (1/2 * 9.8 m/s^2)
time^2 = 10.2s^2
Taking the square root of both sides:
time = √10.2s^2 = 3.2s
(III) The average force exerted by the sand in bringing the body to rest:
We can use the work-energy principle:
Work (W)done = force * distance
Given that the body penetrated 5.0cm of sand, which is 0.05m, and the initial potential energy is 980J, we can calculate the work done by the sand:
Work (W)done = Change in potential energy
W = -980J (negative because work is done against the motion)
Using the equation:
W = force * distance,
-980J = force * 0.05m
force = -980J / 0.05m
force = -19600 N
The negative sign means that the force is acting opposite to the direction of motion, which is expected as the sand brings the body to rest.
Therefore:
(I) The velocity with which the ball hits the sand is 31.3 m/s.
(II) The time taken in falling is 3.2 seconds.
(III) The average force exerted by the sand in bringing the body to rest is 19600 Newtons.