acceleration due to gravity questions

A rock is dropped from a height of 7meters

a) what is the average velocity of the rock during its fall

b) how long does it take the rock to fall

To solve these problems, we need to use the following kinematic equations:

1. v = u + at
2. s = ut + (1/2)at^2
3. v^2 = u^2 + 2as

where v is final velocity, u is initial velocity, a is acceleration, t is time, and s is displacement. In this case, the initial velocity u = 0 (since the rock is dropped), and the acceleration a = -9.81 m/s^2 (due to gravity, acting downward).

a) To find the average velocity of the rock during its fall, we can calculate the final velocity v when the rock reaches the ground, then take the average of the initial and final velocity.

Using equation 3: v^2 = u^2 + 2as, we have:

v^2 = 0 + 2*(-9.81)*(-7)
v^2 = 137.62
v = sqrt(137.62) ≈ -11.73 m/s

(Note that the velocity is negative, meaning it is downward.)

The average velocity during the fall is:

(Initial velocity + Final velocity) / 2 = (0 + (-11.73)) / 2 = -5.865 m/s

b) To find the time it takes the rock to fall, we can use equation 2: s = ut + (1/2)at^2, with s = -7m (downward displacement) and u = 0:

-7 = 0*t + (1/2)*(-9.81)*t^2

Rearranging the equation to solve for t:

t^2 = (2 * -7) / (-9.81)
t^2 ≈ 1.428
t ≈ sqrt(1.428) ≈ 1.195 seconds

So it takes the rock approximately 1.195 seconds to fall.

To answer these questions, we need to calculate the acceleration due to gravity and then use it to solve for the average velocity and time of fall.

The acceleration due to gravity is a constant value, approximately 9.8 m/s². This value represents the rate at which an object falls in a gravitational field.

a) To find the average velocity of the rock during its fall, we can use the equation: average velocity = (final velocity + initial velocity) / 2. Since the rock falls from rest, the initial velocity is 0. The final velocity can be determined using the equation: final velocity = initial velocity + (acceleration × time). In this case, the acceleration due to gravity is acting downwards, so it will be negative since it opposes the motion.

Using these equations, we have: final velocity = 0 + (-9.8 m/s²) × time and average velocity = (0 + [0 + (-9.8 m/s²) × time]) / 2.

b) To calculate the time it takes for the rock to fall, we can use the equation: distance = initial velocity × time + (1/2) × acceleration × time². Since the rock starts from rest, the initial velocity is 0, and the equation simplifies to: distance = (1/2) × acceleration × time².

Now let's actually find the answers.

a) To find the average velocity, we need to know the time it takes for the rock to fall. Let's calculate that first.

For b), the distance is given as 7 meters. Plugging this into the equation, we have: 7 meters = (1/2) × (9.8 m/s²) × time².

Rearranging the equation: time² = (2 × distance) / acceleration.

Substituting the values: time² = (2 × 7 meters) / (9.8 m/s²).

Taking the square root of both sides: time = √((2 × 7 meters) / (9.8 m/s²)).

Evaluating this expression will give us the time it takes for the rock to fall.

Now that we have the time, we can go back to a) and substitute this value into the equation for the final velocity. Using the equation final velocity = 0 + (-9.8 m/s²) × time, we can calculate the final velocity.

Finally, we can substitute the initial and final velocities into the equation for average velocity: average velocity = (0 + [0 + (-9.8 m/s²) × time]) / 2.

Evaluating this expression will give us the average velocity of the rock during its fall.

To find the average velocity and time taken for the rock to fall, we can use the formula for the motion of a falling object under gravity.

a) The average velocity can be found using the formula:

average velocity = (final velocity + initial velocity) / 2

During the fall, since the rock is dropped from rest, the initial velocity is 0 m/s. The final velocity can be calculated using the formula:

final velocity = initial velocity + (acceleration × time)

The acceleration due to gravity is approximately 9.8 m/s^2.

b) The time taken for the rock to fall can be determined using the formula:

time = √(2 × height / acceleration)

Substituting the given values, we can calculate the average velocity and time taken.

a) average velocity = (final velocity + initial velocity) / 2
average velocity = (0 + final velocity) / 2

b) time = √(2 × height / acceleration)
time = √(2 × 7 / 9.8)