A delivery truck accelerates uniformly from rest to a velocity of 8.0 m/ s in 3.0 s. It then travels at a constant speed for 6.0 s. Finally, it accelerates again at a rate of 2.5 m/ s 2 , increasing its speed for 10 s. Determine the truck’s average velocity.
To determine the truck's average velocity, we first need to calculate its total displacement.
First, we need to calculate the displacement during the initial acceleration. We use the equation:
displacement = (initial velocity * time) + (0.5 * acceleration * time^2)
The initial velocity is 0 m/s because the truck starts from rest. The acceleration is given as 2.5 m/s^2, and the time is 3.0 s.
displacement = (0 * 3) + (0.5 * 2.5 * 3^2)
displacement = 0 + 0.5 * 2.5 * 9
displacement = 0 + 11.25
displacement = 11.25 m
The truck's displacement during the initial acceleration is 11.25 m.
Next, we need to calculate the displacement during the constant speed interval. Since the truck travels at a constant speed, the displacement is simply the product of the speed and the time.
speed = 8.0 m/s
time = 6.0 s
displacement = speed * time
displacement = 8.0 * 6.0
displacement = 48 m
The truck's displacement during the constant speed interval is 48 m.
Lastly, we need to calculate the displacement during the final acceleration. We use the same equation as before:
displacement = (initial velocity * time) + (0.5 * acceleration * time^2)
The initial velocity is 8.0 m/s because the truck was already at that velocity at the end of the constant speed interval. The acceleration is given as 2.5 m/s^2, and the time is 10 s.
displacement = (8.0 * 10) + (0.5 * 2.5 * 10^2)
displacement = 80 + 0.5 * 2.5 * 100
displacement = 80 + 0.5 * 2.5 * 100
displacement = 80 + 125
displacement = 205 m
The truck's displacement during the final acceleration is 205 m.
To find the total displacement, we add up the displacements during the three intervals:
total displacement = 11.25 + 48 + 205
total displacement = 264.25 m
The truck's total displacement is 264.25 m.
Finally, we can find the average velocity by dividing the total displacement by the total time.
total time = 3.0 + 6.0 + 10 = 19.0 s
average velocity = total displacement / total time
average velocity = 264.25 / 19.0
average velocity = 13.91 m/s
Therefore, the truck's average velocity is 13.91 m/s.
While undergoing pilot training, a candidate is put in a rocket sled that is initially travelling at 100 km/ h. When the rocket is ignited, the sled accelerates at 30 m/ s 2 . At this rate, how long will it take the rocket sled to travel 500 m down the track?
To find the time it will take for the sled to travel 500 m down the track, we can use the equation:
displacement = (initial velocity * time) + (0.5 * acceleration * time^2)
Given information:
Initial velocity (u) = 100 km/h = 100,000 m/3600 s = 27.78 m/s
Acceleration (a) = 30 m/s^2
Displacement (s) = 500 m
We rearrange the equation to solve for time (t):
500 = (27.78 * t) + (0.5 * 30 * t^2)
500 = 27.78t + 15t^2
15t^2 + 27.78t - 500 = 0
This is a quadratic equation. We can solve it using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 15, b = 27.78, and c = -500.
t = (-27.78 ± √(27.78^2 - 4(15)(-500))) / (2 * 15)
t = (-27.78 ± √(771.8084 + 30000)) / 30
t = (-27.78 ± √(30771.8084)) / 30
t = (-27.78 ± 175.24925) / 30
We ignore the negative value since time cannot be negative:
t = (-27.78 + 175.24925) / 30
t = 147.46925 / 30
t ≈ 4.92 s
Therefore, it will take approximately 4.92 seconds for the rocket sled to travel 500 m down the track.
A parachutist, descending at a constant speed of 17 m/ s, acciden- tally drops his keys, which accelerate downward at 9.8 m/ s 2 . a) Determine the time it takes for the keys to reach the ground if they fall 80 m.
To determine the time it takes for the keys to reach the ground, we can use the kinematic equation:
displacement = (initial velocity * time) + (0.5 * acceleration * time^2)
Given information:
Initial velocity (u) = 0 m/s (since the keys were dropped)
Acceleration (a) = 9.8 m/s^2 (since the keys are accelerating downward due to gravity)
Displacement (s) = -80 m (negative sign indicates the displacement is in the opposite direction of motion)
We rearrange the equation to solve for time (t):
-80 = (0 * t) + (0.5 * 9.8 * t^2)
-80 = 0 + 4.9t^2
4.9t^2 = -80
Since time cannot be negative, we take the positive square root:
t^2 = (-80) / 4.9
t^2 ≈ -16.327
This results in a negative value, which is not physically meaningful. It indicates that the keys would not reach the ground if they were dropped from a height of 80 m with an initial velocity of 0 m/s and an acceleration of 9.8 m/s^2.
However, if the keys were thrown downward with an initial velocity, the equation would be different and provide a valid time for them to reach the ground.
b) What is the final velocity of the keys just before they hit the ground?
To find the final velocity of the keys just before they hit the ground, we can use the equation:
final velocity = initial velocity + (acceleration * time)
Given information:
Initial velocity (u) = 0 m/s (since the keys were dropped)
Acceleration (a) = 9.8 m/s^2 (since the keys are accelerating downward due to gravity)
Displacement (s) = -80 m (negative sign indicates the displacement is in the opposite direction of motion)
Time (t) = ?
As calculated previously, the time it takes for the keys to reach the ground is not possible with the given conditions. However, let's assume that the objects have reached the ground.
Since the displacement is -80 m, the keys have traveled down a distance of 80 m. We consider the downward direction of motion as the positive direction.
Using the equation for displacement:
displacement = initial velocity * time + (0.5 * acceleration * time^2)
-80 = 0 * t + (0.5 * 9.8 * t^2)
Rearranging the equation:
4.9t^2 = -80
Taking the positive value of the square root:
t^2 = -80 / 4.9 ≈ -16.327
This indicates that the keys would not reach the ground based on the given conditions.
Therefore, we cannot determine the final velocity of the keys just before they hit the ground because they would not reach the ground if they were dropped from a height of 80 m with an initial velocity of 0 m/s and an acceleration of 9.8 m/s^2.