To find the force exerted by the sand on the body, we can use the principles of energy conservation and the equations of motion.
First, let's determine the velocity of the body just before it hits the sand. We can do this using the equation for an object in free fall:
\[ v^2 = u^2 + 2gh \]
where:
- \( v \) is the final velocity,
- \( u \) is the initial velocity (which is 0 m/s since the body is falling from rest),
- \( g \) is the acceleration due to gravity (approximately \( 9.81 , \text{m/s}^2 \)),
- \( h \) is the height (3 m).
Substituting the values into the equation:
\[ v^2 = 0 + 2 \times 9.81 \times 3 \] \[ v^2 = 58.86 \] \[ v = \sqrt{58.86} \approx 7.67 , \text{m/s} \]
Now, we can determine the deceleration (or negative acceleration) of the body as it penetrates the sand. Using the following kinematic equation:
\[ v^2 = u^2 + 2as \]
where:
- \( v \) is the final velocity (0 m/s, when the body comes to rest),
- \( u \) is the initial velocity (7.67 m/s),
- \( a \) is the deceleration,
- \( s \) is the distance penetrated into the sand (0.03 m).
Rearranging for \( a \):
\[ 0 = (7.67)^2 + 2a(0.03) \] \[ 0 = 58.86 + 0.06a \] \[ 0.06a = -58.86 \] \[ a = \frac{-58.86}{0.06} \approx -981 , \text{m/s}^2 \]
Now we have the deceleration of the body as it penetrates the sand. To find the force exerted by the sand on the body, we can use Newton's second law of motion:
\[ F = ma \]
where:
- \( m \) is the mass of the body (0.03 kg),
- \( a \) is the total acceleration (which is made up of the deceleration from the penetration into the sand and the acceleration due to gravity).
The total effective acceleration acting on the body when it is in the sand is:
\[ \text{Effective } a = a + g = -981 + 9.81 \approx -971.19 , \text{m/s}^2 \]
Now calculating the force:
\[ F = m \cdot \text{Effective } a \] \[ F = 0.03 \cdot (-971.19) \] \[ F \approx -29.14 , \text{N} \]
The negative sign indicates that the force exerted by the sand is in the opposite direction to the motion of the body. Hence, the magnitude of the force exerted by the sand on the body is approximately:
\[ \text{Force exerted by sand} \approx 29.14 , \text{N} \]
This is the average force exerted by the sand on the body as it comes to rest.
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