A mass m is thrown horizontally at 5.00 m/s off a 20.0m high cliff. find its speed at impact

a)using projectile analysis (20.4m/s)
b) using energy analysis

Vertical component of velocity on impact

= sqrt(2*gh)
=sqrt(2*9.8*20)
=19.8 m/s
horizontal component
=5 m/s
resultant velocity
=sqrt(5²+19.8²)
=20.42 m/s

By energy considerations:
Total kinetic and potential energies at the bottom of the cliff
(1/2)mV²= (1/2)m(5)² + mg(20)
V=sqrt(5²+2g(20))
=20.42 m/s

a) Using projectile analysis:

To find the speed at impact using projectile analysis, we need to consider the horizontal and vertical components separately.

1. Horizontal component:
Since the mass is thrown horizontally, there is no initial velocity in the horizontal direction. The horizontal distance traveled (range) is given by the equation:
range = horizontal velocity x time

The time taken is given by the equation:
time = horizontal distance / horizontal velocity

In this case, the horizontal distance is not given. However, we can assume that the horizontal distance is equal to the range. So, the horizontal velocity remains constant throughout the path.

2. Vertical component:
The vertical motion can be analyzed using the kinematic equation:
vertical displacement = initial velocity x time + (1/2) x acceleration due to gravity x time^2

Since the mass is thrown horizontally, the initial vertical velocity is 0. The displacement is given as 20.0 m (height of the cliff). The acceleration due to gravity is approximately 9.8 m/s^2.

By substituting the values, we can find the time taken for the mass to hit the ground in the vertical direction.

3. Final step:
Once we have the time taken for the vertical motion, we can use that time to find the horizontal distance traveled. Since the horizontal velocity is constant, the horizontal distance is given by the equation:
horizontal distance = horizontal velocity x time

Now that we have the horizontal distance and the time taken, we can find the speed at impact (magnitude of resultant velocity) using the Pythagorean theorem:
resultant velocity = √(horizontal velocity^2 + vertical velocity^2)

By substituting the known values, we can calculate the velocity at impact.

b) Using energy analysis:

We can also find the speed at impact using energy analysis.

1. Potential energy at the top of the cliff:
Potential energy (PE) = mass x gravity x height
PE = m x 9.8 x 20.0

2. At impact, the potential energy would be converted into kinetic energy (KE).
Kinetic energy (KE) = (1/2) x mass x velocity^2
KE = (1/2) x m x velocity^2

By equating the potential energy and kinetic energy at impact, we can solve for the velocity at impact.

m x 9.8 x 20.0 = (1/2) x m x velocity^2

Simplifying the equation, we can solve for the velocity at impact.

So, the speed at impact:
a) using projectile analysis = 20.4 m/s
b) using energy analysis = sqrt(2 x 9.8 x 20.0) ≈ 19.8 m/s

To find the speed at impact, we can approach the problem using two different methods: projectile analysis and energy analysis.

a) Projectile Analysis:
In projectile analysis, we can split the motion into horizontal and vertical components.

- Horizontal Component: Since the mass is thrown horizontally, its initial horizontal velocity (Vx) remains constant throughout the motion. Therefore, Vx = 5.00 m/s.

- Vertical Component: The mass is thrown vertically downward due to gravity. The initial vertical velocity (Vy) is 0 m/s since there is no initial vertical motion. The acceleration due to gravity (g) is -9.8 m/s^2 (negative because it acts downward). The height (h) is given as 20.0 m.
Using the equation of motion:
h = Vy * t + (1/2) * g * t^2

Substituting the initial conditions:
20.0 m = 0 * t + (1/2) * (-9.8 m/s^2) * t^2
20.0 m = -4.9 m/s^2 * t^2

Solving for time (t):
t^2 = 20.0 m / (-4.9 m/s^2)
t^2 = 4.0816 s^2
t = √4.0816 s^2
t = 2.02 s (taking the positive square root as time cannot be negative)

Now, using the time (t), we can find the final vertical velocity (Vfy) using the equation:
Vfy = Vy + g * t
Vfy = 0 m/s + (-9.8 m/s^2) * 2.02 s
Vfy = -19.8 m/s

Finally, we can find the magnitude of the final velocity (Vf) by combining the horizontal and vertical components using the Pythagorean theorem:
Vf = √(Vx^2 + Vfy^2)
Vf = √((5.00 m/s)^2 + (-19.8 m/s)^2)
Vf = √(25.00 m^2/s^2 + 392.04 m^2/s^2)
Vf = √(417.04 m^2/s^2)
Vf = 20.4 m/s (approximately)

Therefore, the speed at impact, using projectile analysis, is approximately 20.4 m/s.

b) Energy Analysis:
In energy analysis, we can analyze the change in potential energy to find the speed at impact.

The potential energy at the top of the cliff (PE_initial) is given by:
PE_initial = m * g * h
where mass (m) is given and g is the acceleration due to gravity (9.8 m/s^2), and h is the height (20.0 m).

The kinetic energy at the bottom of the cliff (KE_final) is given by:
KE_final = (1/2) * m * Vf^2
where Vf is the final velocity (which we need to find).

According to the principle of conservation of energy, the potential energy at the top is converted into kinetic energy at the bottom:
PE_initial = KE_final
m * g * h = (1/2) * m * Vf^2
Canceling mass (m) on both sides:
g * h = (1/2) * Vf^2
Rearranging the equation to solve for Vf:
Vf^2 = 2 * g * h
Taking the square root of both sides to find the magnitude of Vf:
Vf = √(2 * g * h)
Vf = √(2 * 9.8 m/s^2 * 20.0 m)
Vf = √(392 m^2/s^2)
Vf = 19.8 m/s (approximately)

Therefore, the speed at impact, using energy analysis, is approximately 19.8 m/s.