Using Equation Substitution: Videos & Practice Problems

In projectile motion, when faced with two unknowns, such as initial velocity and angle, equation substitution is essential. For example, when a soccer ball is kicked from a hill, knowing the time of flight and horizontal distance allows us to set up equations for both the x and y axes. By expressing one variable in terms of another, we can solve for unknowns like initial velocity and angle using trigonometric relationships. This method streamlines problem-solving in complex scenarios, reinforcing the importance of understanding vector decomposition and the equations of motion.

Solving Projectile Motion Using Equation Substitution

In projectile motion problems, it's common to encounter situations where you need to analyze both the horizontal (x) and vertical (y) components of motion. When faced with unknown variables, such as time, initial velocity, or angle, it's often necessary to switch between the two axes to find a solution. For instance, if you have a horizontal launch and need to calculate the horizontal displacement (\(\Delta x\)), you can start with the equation:

\(\Delta x = v_x \cdot t\)

Here, \(v_x\) is the horizontal velocity, and \(t\) is the time of flight. If time is unknown, you can derive it from the vertical motion equations. This method allows you to find the necessary variables by leveraging the relationships between the two axes.

Consider a scenario where a soccer ball is kicked from a hill, remaining in the air for 4.5 seconds and landing 45 meters away. To analyze this, you can break down the motion into intervals, focusing on the total time and horizontal displacement. The horizontal motion can be expressed as:

\(45 = v_0 \cdot \cos(\theta) \cdot 4.5\)

From this, you can isolate \(v_0\) as:

\(v_0 = \frac{10}{\cos(\theta)}\

However, this leaves you with two unknowns: \(v_0\) and \(\theta\). To resolve this, you can turn to the vertical motion. The vertical displacement (\(\Delta y\)) can be calculated using:

\(\Delta y = v_{0y} \cdot t + \frac{1}{2} a_y t^2\

In this case, \(a_y\) is the acceleration due to gravity, which is approximately \(-9.8 \, \text{m/s}^2\). By substituting \(v_{0y} = v_0 \cdot \sin(\theta)\) into the equation, you can express the vertical motion as:

\(-5 = v_0 \cdot \sin(\theta) \cdot 4.5 - \frac{1}{2} \cdot 9.8 \cdot (4.5)^2\

Rearranging gives:

21 = \(v_0 \cdot \sin(\theta)\

Now, you have two equations with two unknowns. To solve for one variable, you can use substitution. From the first equation, substitute \(v_0\) into the second equation:

21 = \(\frac{10}{\cos(\theta)} \cdot \sin(\theta)\

This simplifies to:

21 = 10 \cdot \tan(\theta)

From which you can find:

\(\theta = \tan^{-1}(2.1)\

Calculating this gives an angle of approximately \(64.5^\circ\). With \(\theta\) known, you can substitute back to find \(v_0\) using either of the original equations. This method of equation substitution is particularly useful when dealing with multiple unknowns in projectile motion problems, allowing you to systematically solve for each variable.

A ball is kicked at a 45° angle from the ground. It hits the wall of a building 30 m away, 10m up from the ground. What was the ball's initial velocity?

• A. 17.7 m/s
• B. 3.2 m/s
• C. 21 m/s
• D. 441 m/s

Answer: (C)

Here’s what students ask on this topic:

How do you use equation substitution in projectile motion problems?

In projectile motion problems, equation substitution is used when you have two unknowns, such as initial velocity and angle. First, set up equations for both the x and y axes. For example, if you know the time of flight and horizontal distance, you can express the horizontal displacement as $\mathrm{\Delta x}=v$₀ cos θ t. Similarly, for the vertical displacement, use $\mathrm{\Delta y}=v$₀ sin θ t - 0.5 g t². Solve one equation for one variable, then substitute it into the other equation to find the unknowns. This method simplifies solving complex scenarios by breaking them into manageable parts.

What is the importance of vector decomposition in solving projectile motion problems?

Vector decomposition is crucial in solving projectile motion problems because it allows you to break down the initial velocity into horizontal and vertical components. This is essential for setting up equations for each axis. For instance, the initial velocity $v$₀ can be decomposed into $v$₀ cos θ for the x-axis and $v$₀ sin θ for the y-axis. This decomposition helps in applying the equations of motion separately to each axis, making it easier to solve for unknowns like time, range, and maximum height.

Can you explain a step-by-step method to solve for initial velocity and angle in a projectile motion problem using equation substitution?

Sure! Here’s a step-by-step method:

1. Identify the known values (e.g., time of flight, horizontal distance).
2. Set up the horizontal motion equation: $\mathrm{\Delta x}=v$₀ cos θ t.
3. Set up the vertical motion equation: $\mathrm{\Delta y}=v$₀ sin θ t - 0.5 g t².
4. Solve one equation for one variable (e.g., $v$₀ = Δx / (cos θ t)).
5. Substitute this expression into the other equation.
6. Solve for the remaining variable (e.g., $\theta =\tan -1(\mathrm{\Delta y}/\; (\mathrm{\Delta x}-\; 0.5gt2))$).
7. Use the found value to solve for the initial velocity.

This method ensures you systematically solve for both unknowns.

What are common mistakes to avoid when using equation substitution in physics problems?

Common mistakes to avoid include:

1. Not correctly decomposing vectors into their components, leading to incorrect equations.
2. Forgetting to keep track of units, which can result in errors in calculations.
3. Not properly isolating one variable before substituting it into another equation, which can complicate the solution process.
4. Ignoring the signs of variables, especially in vertical motion where gravity is involved.
5. Overlooking the need to use trigonometric identities correctly, such as $\tan \theta =\sin \theta /\cos \theta$.

Being meticulous with these aspects ensures accurate and efficient problem-solving.

How does understanding the equations of motion help in using equation substitution effectively?

Understanding the equations of motion is fundamental for effective equation substitution because it allows you to set up accurate equations for both the x and y axes. The key equations are:

• Horizontal motion: $\mathrm{\Delta x}=v$₀ cos θ t
• Vertical motion: $\mathrm{\Delta y}=v$₀ sin θ t - 0.5 g t²

By knowing these equations, you can express one variable in terms of another and substitute it into the second equation. This method simplifies solving for unknowns like initial velocity and angle, making the problem more manageable.