Question

An object is thrown upward with a speed of \(28.0 \mathrm{~m} / \mathrm{s}\). How high above the projection point is it after 1.00 s?

Short answer

Answer: The object is 23.095 m above the projection point.

Step-by-step solution

Identify the parameters

Given the initial velocity \(v_0 = 28.0 \mathrm{~m/s}\) and the time \(t = 1.00 \mathrm{~s}\). The acceleration due to gravity is \(g = 9.81 \mathrm{~m/s^2}\), and it acts opposite to the initial velocity.

Choose the kinematic equation

We will use the following kinematic equation, which relates displacement, initial velocity, time, and acceleration: \(y = v_0t - \frac{1}{2}gt^2\). In this equation, \(y\) is the vertical displacement or height above the projection point.

Plug in the values and calculate the height

Insert the given values into the equation: \(y = (28.0 \mathrm{~m/s})(1.00 \mathrm{~s}) - \frac{1}{2}(9.81 \mathrm{~m/s^2})(1.00 \mathrm{~s})^2\). Now, perform the calculations: \(y = 28.0 \mathrm{~m} - 4.905 \mathrm{~m}\) \(y = 23.095 \mathrm{~m}\)

Report the final answer

The object is \(23.095 \mathrm{~m}\) above the projection point after \(1.00 \mathrm{~s}\).

Key concepts

Initial Velocity

In kinematics, the initial velocity is a crucial parameter that describes the speed and direction an object travels when first observed or set into motion. It is typically denoted as \(v_0\). In the exercise, our object is thrown upwards at an initial velocity of \(28.0 \mathrm{~m/s}\). This means that at the start of its motion, it moves upwards with this speed.

The initial velocity helps determine how far and how long an object will continue to ascend before gravity slows it down. Understanding the initial velocity allows us to predict the object's future position on its trajectory. It serves as the starting point in kinematic equations, allowing us to calculate other aspects of motion like time, displacement, and final velocity.

• Sets the motion context.
• Defines the initial direction and speed.
• Essential for solving kinematics problems.

Acceleration Due to Gravity

Acceleration due to gravity, denoted as \(g\), is a fundamental concept in kinematics, especially when dealing with vertical motion. It describes how the velocity of an object changes due to the gravitational pull of the Earth. The standard value of \(g\) is approximately \(9.81 \mathrm{~m/s^2}\). In our exercise, this acceleration acts downward, opposing the object's initial upward velocity.

Gravity's acceleration is what eventually stops an object moving upwards and then pulls it back down. It's always directed towards the center of the Earth, influencing the object's overall motion. By understanding this, we can better predict how far an object will move upwards before it starts descending.

• Constant and downward-directed.
• Key in determining the maximum height.
• Affects all objects regardless of their masses.

Kinematic Equations

Kinematic equations are mathematical formulas used to describe the motion of objects without considering the forces that cause the motion. These equations relate the four kinematic variables: displacement, initial velocity, time, and acceleration.

In the given exercise, we use the equation \(y = v_0t - \frac{1}{2}gt^2\) to find the vertical displacement. This equation is essential as it combines both the effects of the initial velocity and the gravitational pull on the object over time.

• Relate motion variables.
• Work without external forces.
• Include time, velocity, and displacement.

Vertical Displacement

Vertical displacement, denoted by \(y\), represents the height an object reaches relative to its initial starting point. It is the difference in height from where the object was projected to where it is observed after a certain time.

In our exercise, the calculation of vertical displacement gives us \(23.095 \mathrm{~m}\), indicating that the object is \(23.095\) meters above its starting point after \(1\) second. This displacement is determined using both the initial upward motion and the opposing force of gravity.

• Holds key information about height achieved.
• Calculated using kinematic equations.
• Critical for understanding motion trajectory.