A go-cart has an initial speed of 23.4 m/sec. Fifteen seconds later the go-cart has a final speed of 36.8 m/sec. What is the magnitude of the go-cart's displacement?
average speed= 1/2 (Vf+Vi)
distance=averagespeed*time
To find the displacement of the go-cart, we can use the equation:
displacement = final position - initial position
Since we do not have any information about the positions, we will use the equation a = (vf - vi)/(t), where a is acceleration, vf is final velocity, vi is initial velocity, and t is time.
First, let's calculate the acceleration of the go-cart using this equation:
a = (vf - vi)/(t)
= (36.8 m/s - 23.4 m/s)/(15 s)
Simplifying the equation:
a = (13.4 m/s)/(15 s)
= 0.8933 m/s^2
Now, we can use the equation v^2 = u^2 + 2as, where v is final velocity, u is initial velocity, a is acceleration, and s is the displacement.
To find the displacement, we rearrange the equation:
s = (v^2 - u^2)/(2a)
= (36.8 m/s)^2 - (23.4 m/s)^2 / (2 * 0.8933 m/s^2)
= 1356.64 m^2/s^2 - 547.56 m^2/s^2 / 1.7866 m/s^2
≈ 170.45 m
Therefore, the magnitude of the go-cart's displacement is approximately 170.45 meters.
To find the magnitude of the go-cart's displacement, we need to determine the change in its position from the initial to the final position. This can be calculated using the formula:
Displacement = Final Position - Initial Position
In this case, we are given the initial speed and the time elapsed, but we don't have the distances. We can calculate the initial position and final position using the kinematic equations of motion.
The formula to find the position (distance) traveled during constant acceleration is:
Position = Initial Velocity * Time + (1/2) * Acceleration * Time^2
Since the go-cart is moving with a constant acceleration (which we don't know), we can apply this formula to both the initial and final positions to find the initial and final displacements.
Let's calculate the initial displacement:
Initial Velocity = 23.4 m/sec
Time = 0 seconds (since the initial position is the starting point)
Using the formula, we have:
Initial Displacement = Initial Velocity * Time + (1/2) * Acceleration * Time^2
Initial Displacement = 23.4 * 0 + (1/2) * Acceleration * 0^2
Initial Displacement = 0
Now, let's calculate the final displacement:
Initial Velocity = 23.4 m/sec
Time = 15 seconds
Using the formula, we have:
Final Displacement = Initial Velocity * Time + (1/2) * Acceleration * Time^2
Final Displacement = 23.4 * 15 + (1/2) * Acceleration * 15^2
Final Displacement = 351 + (1/2) * Acceleration * 225
Final Displacement = 351 + 112.5 * Acceleration
Now, to find the acceleration, we can use the following formula derived from the given information:
Final Velocity = Initial Velocity + Acceleration * Time
Substituting the known values:
36.8 = 23.4 + Acceleration * 15
36.8 - 23.4 = 15 * Acceleration
13.4 = 15 * Acceleration
Acceleration = 13.4 / 15
Acceleration = 0.893 m/sec^2
Now, we can substitute the value of acceleration back into the final displacement equation:
Final Displacement = 351 + 112.5 * 0.893
Final Displacement = 351 + 100.23375
Final Displacement = 451.23375
The magnitude of the go-cart's displacement is approximately 451.23 meters.