Displacement
Displacement is a vector quantity that describes the overall change in position of an object. It is represented by the symbol s or Delta x or Delta r and is measured in meters (m). It is crucial to note that displacement not only considers the change in position but also the direction of motion.
Mathematical Representation
s = sf - si
Where:
- s = Displacement
- sf = Final position
- si = Initial position
Detailed Explanation
Displacement is not the same as distance. While distance is a scalar quantity representing the total path length travelled, displacement is concerned with the direct straight-line change in position from the initial to the final point. It is possible for an object to travel a considerable distance and yet have a displacement of zero if it returns to its starting point.
Example Question 1
A runner jogs 100 meters towards the east and then 150 meters towards the west. Calculate the total displacement of the runner.
Solution
Since displacement is a vector quantity, we must consider the direction in the calculation. The total displacement s is given by:
s = seast + swest
Where:
- seast = 100 m
- swest = -150 m (west is considered negative)
s = 100 m - 150 m = -50 m
The negative sign indicates that the net displacement is 50 meters towards the west.
Velocity
Velocity, denoted by v, is the rate of change of displacement with respect to time. It is a vector quantity, meaning it has both magnitude and direction, and is measured in meters per second (m/s).
Mathematical Representation
v = (Delta s) / (Delta t)
Where:
- Delta s = Change in displacement
- Delta t = Change in time
Detailed Explanation
Velocity can be constant or variable. When an object moves with constant velocity, it covers equal displacements in equal intervals of time. However, if the velocity is changing, the object is said to be accelerating, which brings us to our next concept later on.
Example Question 2
A cyclist travels a displacement of 200 meters towards the north in 25 seconds. Determine its velocity.
Solution
Using the formula for velocity:
v = (Delta s) / (Delta t)
v = (200 m) / (25 s)
v = 8 m/s north
Acceleration
Acceleration, symbolized by a, is the rate of change of velocity with respect to time. It is also a vector quantity and is measured in meters per second squared (m/s²).
Mathematical Representation
a = (Delta v) / (Delta t)
Where:
- Delta v = Change in velocity
- Delta t = Change in time
Detailed Explanation
Acceleration can be positive, negative, or zero. Positive acceleration (often referred to simply as 'acceleration') indicates an increase in velocity, while negative acceleration (often referred to as 'deceleration') indicates a decrease in velocity. Zero acceleration means the velocity is constant.
Example Question 3
A skateboarder increases its velocity from 5 m/s to 15 m/s over a time interval of 2 seconds. Calculate the acceleration.
Solution
Using the formula for acceleration:
a = (Delta v) / (Delta t)
a = (15 m/s - 5 m/s) / (2 s)
a = (10 m/s) / (2 s)
a = 5 m/s^2
Relationships Among Displacement, Velocity, and Acceleration
The relationships among displacement, velocity, and acceleration are vital for solving kinematics problems. The equations of motion, which are often used, are:
- v = u + at
- s = ut + 0.5 * at2
- v2 = u2 + 2as
Where:
- u = Initial velocity
- v = Final velocity
- a = Acceleration
- s = Displacement
- t = Time
Example Question 4
A train starts from rest and accelerates uniformly at 2 m/s². Find its velocity after 7 seconds.
Solution
Using the first equation of motion:
v = u + at
Since the train starts from rest, u = 0. Substituting the values we have:
v = 0 m/s + (2 m/s2 * 7 s)
v = 14 m/s
FAQ
An object can have a constant speed but not a constant velocity because speed is a scalar quantity, only considering the magnitude, while velocity is a vector quantity, considering both magnitude and direction. If an object is moving at a constant speed but changing its direction, it has a constant speed but not a constant velocity. A classic example is an object moving in a circular path at a steady speed. The magnitude of the speed remains constant, but the direction of motion is continuously changing, hence, the velocity, which accounts for both speed and direction, is not constant.
Instantaneous velocity refers to the velocity of an object at a specific point in time or at a specific point in its path. It is the rate of change of displacement with respect to time, considering an infinitesimally small time interval. On the other hand, average velocity is calculated over a finite time interval and is the total displacement divided by the total time taken. Instantaneous velocity is represented as the slope of the tangent to the displacement-time graph at a particular point, while average velocity is represented as the slope of the chord connecting two points on the displacement-time graph. Both are vector quantities and have both magnitude and direction.
Yes, an object can be in motion even if its velocity is zero at a particular instant. A simple example to illustrate this is a ball thrown vertically upwards. At the highest point of its trajectory, its velocity is momentarily zero, but it is still in motion since it is in the process of changing its position. The velocity is zero at the peak because, for that instant, the ball stops moving upwards before it starts its descent back down. However, the ball is still considered to be in motion over the duration of its upward and downward journey because its position changes with respect to time.
The area under a velocity-time (v-t) graph represents the displacement of an object over a specified time interval. This is because the integral (or the area under the curve) of velocity with respect to time gives displacement. If the velocity is constant, the v-t graph is a rectangle, and the displacement (s) can be found by multiplying velocity (v) and time (t) (s = v * t). If the velocity is changing, the v-t graph may be a different shape, and the displacement is found by calculating the area under the curve. If the graph is below the time axis (indicating negative velocity), the displacement is negative, showing that the object has moved in the opposite direction.
Air resistance, or drag, is a force that opposes the motion of an object through a fluid (like air). When an object is in free fall, two forces are acting upon it: gravitational force pulling it downwards and air resistance pushing it upwards. As an object accelerates downwards due to gravity, the air resistance acting against it also increases. This eventually leads to a point where the gravitational force pulling the object downwards equals the air resistance force pushing upwards. At this point, the net force acting on the object becomes zero, and it stops accelerating and continues to fall at a constant velocity, known as terminal velocity. The impact of air resistance is especially noticeable in objects with a large surface area, like a parachute, slowing down the descent significantly.
Practice Questions
The acceleration (a) can be found using the formula: a = (v - u) / t Where:
- v = final velocity = 60 m/s
- u = initial velocity = 0 m/s (since the car starts from rest)
- t = time = 30 s
Substituting the values, we get: a = (60 m/s - 0 m/s) / 30 s a = 2 m/s²
To find the displacement (s), we can use the formula: s = ut + 0.5 * a * t²
Substituting the values, we get: s = (0 m/s * 30 s) + 0.5 * (2 m/s² * (30 s)²) s = 0 + 0.5 * (2 m/s² * 900 s²) s = 900 m
The maximum height (H) is reached when the final velocity (v) of the ball is 0 m/s. We can use the following equation of motion to find the maximum height: v² = u² + 2as Where:
- v = final velocity = 0 m/s (at maximum height)
- u = initial velocity = 20 m/s
- a = acceleration = -9.8 m/s² (negative since gravity acts downwards)
- s = displacement = H (maximum height)
Rearranging the formula to find H, we get: H = (v² - u²) / (2a)
Substituting the values, we get: H = (0 - (20 m/s)²) / (2 * -9.8 m/s²) H = (-400 m²/s²) / (-19.6 m/s²) H = 20.41 m
The ball reaches a maximum height of 20.41 m.
Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.