Find Acceleration

 

What is the range?

A projectile is launched at an angle of 10 and lands 30 s later at the same height as it was launched.
What is the range?
Input your answer in m in scientific e-notation using 3 significant figures.

Solution

θ = 10°
t = 30s
x = ?

v cos(θ) t = x
gt = 2vsin(θ)

v=gt /(2sin(θ))

gt /(2sin(θ)) cos(θ) t = x

gt

²

 cot(θ) /2 = x

x = gt² cot(θ) /2
θ = 10°
t = 30s

g=9.8 m/s²

WolframAlpha: 
 find x: θ = 10 degrees, t = 30, g=9.8, x = g*t² cot(θ) /2 

https://www.wolframalpha.com/input?i=+find+x%3A+%CE%B8+%3D+10+degrees%2C+t+%3D+30%2C+g%3D9.8%2C+x+%3D+g*t%C2%B2+cot%28%CE%B8%29+%2F2 


Google Search:

9.8 m/s^2*(30s)^2*cot(10deg)/2 in m 

How long will it take a log to float from the dam to the grove?

The beavers noticed that swimming from the dam to the grove along the river takes time t₁, and swimming back along the river from the grove to the dam takes time t₂, and t₁ < t₂.

How long (t=?) will it take a log to float from the dam to the grove?



Solution



t₁ (V_s + V_r)  =  L

t₂ (V_s - V_r)  =  L

t  V_r  =  L


 (V_s + V_r)  =  L/t₁

 (V_s - V_r)  =  L/t₂

(V_s + V_r) - (V_s - V_r)   =   L/t₁ - L/t₂ 

2 V_r    =     L(1/t₁ - 1/t₂)

V_r   =    L(1/t₁ - 1/t₂)/2

t =  L/ V_r  = L /  ( L(1/t₁ - 1/t₂)/2)     =  2t₁ t₂ / (t₂ - t₁)

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2 Key Equations

 

Accelerations a₁ and a₂ of two separate objects

Problem:

The same net force causes accelerations a₁ and a₂ of two separate objects.
What will be the acceleration if these bodies are fastened together and the resultant force mentioned above is applied to this system?


Solution:

To find the acceleration of two objects when connected and subjected to the same net force, let's apply the principle F=ma correctly:

Given:

• $�$
• F is the net force applied to both objects when they are connected.
• ${�}_1$ and ${�}_2$ are the accelerations of the two objects when the force is applied separately to each.
• ${�}_1$ and ${�}_2$ are the masses of the two objects, respectively.

From Newton's second law, when the force $�$ is applied to each object separately: $�={�}_1{�}_1$ $�={�}_2{�}_2$

These equations give us the individual masses of the objects as: ${�}_1=\frac{�}{{�}_1}$ ${�}_2=\frac{�}{{�}_2}$

When the objects are connected and the same force $�$ is applied, the total mass is ${�}_1+{�}_2$. The acceleration $�$ of this combined system under the force $�$ is given by: $�=({�}_1+{�}_2)�$

Substituting ${�}_1$ and ${�}_2$ from above into the equation: $�=(\frac{�}{{�}_1}+\frac{�}{{�}_2})�$

Solving for $�$, we get: $�=\frac{�}{(\frac{�}{{�}_1}+\frac{�}{{�}_2})}$

This simplifies to: $�=\frac{1}{(\frac{1}{{�}_1}+\frac{1}{{�}_2})}$

So, the acceleration of the combined system is determined by the reciprocal of the sum of the reciprocals of the individual accelerations. This formula shows how the combined acceleration relates to the individual accelerations ${�}_1$ and ${�}_2$.