Question

You throw a ball from a cliff with an initial velocity of \(15 \mathrm{~m} / \mathrm{s}\) at an angle of \(20^{\circ}\) below the horizontal. Find \((a)\) its horizontal displacement and \((b)\) its vertical displacement \(2.3\) s later

Short answer

The horizontal displacement of the ball after 2.3 seconds is approximately 33.02 meters, and the vertical displacement is approximately -16.32 meters.

Step-by-step solution

Find the Horizontal and Vertical Component of Initial Velocity

First, remember that the initial velocity has both a horizontal and vertical component. For an angle below the horizontal, the horizontal component \(v_{ix}\) can be calculated as \(v_{i} * cos(\theta)\) where \(v_{i}=15 \mathrm{m/s}\) is the magnitude of initial velocity and \(\theta = 20^{\circ}\). Similarly, the vertical component \(v_{iy}\) can be calculated as \(v_{i} * sin(\theta)\). Be mindful that because the angle is below the horizontal the vertical component will be negative, as it directs downwards.

Calculate Horizontal Displacement

In projectile motion, horizontal displacement is given by \(x = v_{ix} * t\), where \(x\) is the horizontal displacement, \(v_{ix}\) is the horizontal component of initial velocity and \(t=2.3s\) is the time. Substitute the values into the equation to get the horizontal displacement.

Calculate Vertical Displacement

In projectile motion, vertical displacement is given by \(y = v_{iy} * t + 0.5 * g * t^{2}\), where \(y\) is the vertical displacement, \(v_{iy}\) is vertical component of initial velocity, \(t=2.3s\) is the time and \(g=9.8 \mathrm{m/s^2}\) is the acceleration due to gravity. Substitute the values into the equation to get the vertical displacement. Keep in mind that the displacement is negative as the motion is downward.

Key concepts

Horizontal Displacement

Understanding horizontal displacement in projectile motion is crucial in physics. In simple terms, horizontal displacement refers to how far an object travels along the horizontal axis during its motion.

When you throw an object like a ball, it travels both in a horizontal and a vertical direction. The horizontal movement is not affected by gravity, which means the object would continue moving horizontally at a constant speed if there were no other forces acting upon it.

To calculate the horizontal displacement, you need to know:

• The horizontal component of the initial velocity (\(v_{ix}\)). This is affected by the angle and speed at which you throw the object.
• The time the object is in motion (\(t\)).

The formula to find horizontal displacement is given by \(x = v_{ix} * t\).
This means that if you know how fast the object is moving in the horizontal direction and for how long, you can easily find out how far it has moved horizontally.

Vertical Displacement

Vertical displacement in projectile motion can sometimes be a bit more complex than horizontal motion because gravity plays a significant role.

When we talk about vertical displacement, we mean how high or low the projectile has moved compared to its starting point. In this exercise, because the ball was thrown downward, the vertical displacement will be negative, indicating a downward movement.

Here's what you need to calculate vertical displacement:

• The vertical component of the initial velocity (\(v_{iy}\)).
• Time of flight (\(t\)). Again, this tells you how long the object has been moving.
• Acceleration due to gravity (\(g\)), which is constant at approximately \(9.8 \mathrm{m/s^2}\).

The formula for vertical displacement incorporates these factors: \[ y = v_{iy} * t + 0.5 * g * t^{2} \]Considering that the angle is below the horizontal, the initial vertical velocity is negative, indicating that the motion begins downward. This equation helps us understand both how far and in what direction the object moves vertically.

Initial Velocity Components

To understand the motion of a projectile like a thrown ball, breaking down its initial velocity into two components is essential. This helps in analyzing how the object moves both horizontally and vertically.

The initial velocity is given in magnitude and direction. However, in physics, it is often helpful to break this vector into horizontal and vertical components:

• Horizontal Component (\(v_{ix}\)): This component can be calculated using the cosine of the angle, \(v_{ix} = v_{i} \cdot \cos(\theta)\). It represents how fast and in what direction the object travels horizontally. Because the angle is below horizontal, it's usually measured from the positive horizontal axis.
• Vertical Component (\(v_{iy}\)): This component uses the sine of the angle, \(v_{iy} = v_{i} \cdot \sin(\theta)\). In our exercise, the vertical component is negative because the direction is downward due to the angle being below the horizontal.

Understanding these components allows you to predict both how far and how high (or low) a projectile will travel within a given time frame. Once you have these components, you can utilize them to find the horizontal and vertical displacements.