Question

A stone is thrown upward, from ground level, with an initial velocity of \(10.0 \mathrm{~m} / \mathrm{s}\). a) What is the velocity of the stone after 0.50 s? b) How high above ground level is the stone after 0.50 s?

Short answer

Answer: The velocity of the stone after 0.50 s is 5.095 m/s, and its height above ground level is 3.77375 m.

Step-by-step solution

Identify the relevant variables and kinematic formula.

We need to find the final velocity (v) of the stone after 0.50 s with an initial velocity (u) of 10.0 m/s. The acceleration due to gravity (a) is -9.81 m/s², and the time (t) is 0.50 s. The kinematic formula we need is: v = u + at

Plug in the known values and solve for the final velocity.

Input the given values into the kinematic formula: v = 10.0 m/s + (-9.81 m/s²)(0.50 s) Now, solve for v: v = 10.0 m/s - 4.905 m/s v = 5.095 m/s So, after 0.50 s, the velocity of the stone is 5.095 m/s. b) Finding the height above ground level after 0.50 s.

Identify the relevant variables and kinematic formula.

We need to find the displacement (s) of the stone from the ground after 0.50 s with an initial velocity (u) of 10.0 m/s. The acceleration due to gravity (a) is -9.81 m/s², and the time (t) is 0.50 s. The kinematic formula we need is: s = ut + 0.5at²

Plug in the known values and solve for the displacement.

Input the given values into the kinematic formula: s = (10.0 m/s)(0.50 s) + 0.5(-9.81 m/s²)(0.50 s)² Now, solve for s: s = 5.0 m - 1.22625 m s = 3.77375 m So, after 0.50 s, the stone is 3.77375 m above the ground level.

Key concepts

Initial Velocity

Initial velocity is the speed and direction at which an object starts moving. It is often denoted by the symbol \( u \). In our exercise, the stone is thrown upward with an initial velocity of \( 10.0 \, \mathrm{m/s} \).

• This initial velocity determines how fast and in what direction the stone will start its motion.
• The positive value of \( 10.0 \, \mathrm{m/s} \) indicates that the stone is moving upward, opposite the direction of gravity.

Understanding initial velocity is crucial because it is the starting point for calculating other kinematic equations, such as final velocity and displacement.

Final Velocity

Final velocity refers to the speed and direction of an object at the end of a given time interval. It is often represented by the letter \( v \). After 0.50 seconds, the stone's velocity was affected by gravity, resulting in a value of \( 5.095 \, \mathrm{m/s} \).

• The calculation for final velocity uses the equation: \[ v = u + at \] where \( u \) is initial velocity, \( a \) is acceleration, and \( t \) is time.
• In this problem, gravity acts as the acceleration, which is \( -9.81 \, \mathrm{m/s^2} \).
• This negative sign indicates that gravity slows the stone's upward motion.

With this understanding, the final velocity becomes: \( v = 10.0 \, \mathrm{m/s} - 4.905 \, \mathrm{m/s} \), leading to \( v = 5.095 \, \mathrm{m/s} \).
Thus, understanding this concept helps us to predict the velocity changes over time.

Acceleration Due to Gravity

Acceleration due to gravity is a constant force that pulls objects toward the Earth’s center. It is a crucial factor in motion equations, impacting objects thrown, dropped, or moving in air.

• On Earth, this acceleration is \( -9.81 \, \mathrm{m/s^2} \), often denoted by \( g \).
• The negative sign indicates that it acts downward, opposite to any initial upward motion.

In our problem, the stone experiences this acceleration, affecting both its final velocity and displacement. It continuously reduces the stone’s upward velocity until the stone begins to fall back down.
The understanding of this concept is necessary for calculating motions accurately.

Displacement

Displacement is the overall change in position of an object and can be calculated with the kinematic equation: \[ s = ut + \frac{1}{2}at^2 \]. It gives a clear picture of how far and in which direction an object has moved from its starting point.

• In the example of the stone thrown upwards, displacement shows not just how much the stone has traveled, but also that it is above its starting point after 0.50 seconds.
• Calculating the stone's displacement involves using initial velocity \( u = 10.0 \, \mathrm{m/s} \), time \( t = 0.50 \, \mathrm{s} \), and acceleration \( a = -9.81 \, \mathrm{m/s^2} \).
• Applying these into the equation:
  \( s = (10.0 \, \mathrm{m/s})(0.50 \, \mathrm{s}) + 0.5(-9.81 \, \mathrm{m/s^2})(0.50 \, \mathrm{s})^2 \) results in a displacement of \( 3.77375 \, \mathrm{m} \).

Hence, after half a second, the stone has moved approximately 3.77 meters above its original position.
Displacement helps us measure this net change in location, an essential part of kinematics.