Kinematics

To determine the instantaneous velocity and acceleration of the body, we need to perform the following steps, assuming that 'PT squared' refers to 'P times T squared', where P is a constant and T is time:

1. Define the displacement equation:

Given: Displacement, s(t) = PT² + 40T + 7

2. Calculate the instantaneous velocity:

Instantaneous velocity, v(t), is the first derivative of displacement with respect to time (T).

v(t) = ds/dt = d(PT² + 40T + 7)/dt

v(t) = 2PT + 40

3. Calculate the instantaneous acceleration:

Instantaneous acceleration, a(t), is the first derivative of velocity with respect to time (T), or the second derivative of displacement with respect to time.

a(t) = dv/dt = d(2PT + 40)/dt

a(t) = 2P

Therefore:

• Instantaneous Velocity: v(t) = 2PT + 40
• Instantaneous Acceleration: a(t) = 2P

To find the speed, we will use the formula: Speed = Distance / Time

Given:

• Distance = 1440 km
• Time = 2 hours

First, calculate the speed in km/h:

Speed = 1440 km / 2 hours = 720 km/h

Now, convert the speed from km/h to m/s:

1 km = 1000 meters

1 hour = 3600 seconds

Therefore, to convert km/h to m/s, multiply by 1000/3600 (which simplifies to 5/18):

Speed in m/s = 720 km/h * (5/18)

Speed in m/s = 720 * (5/18) = 200 m/s

Therefore, the person's speed is 720 km/h or 200 m/s.

The first of Newton's equations of motion relates final velocity, initial velocity, acceleration, and time.

The equation is:

• Where:
  • v = final velocity
  • u = initial velocity
  • a = acceleration
  • t = time

This equation directly gives the relation between velocity (both final and initial) and time, under constant acceleration.