1)
The acceleration of the car is the rate of change of velocity of the car; it can be calculated as:
where
u is the initial velocity
v is the final velocity
t is the time taken for the velocity of the car to change from u to v
In this problem, for this car we have:
u = 37.1 m/s
v = 29.8 m/s
t = 3 s
So, the acceleration is:
2)
The work done in lifting the box is equal to the potential energy transferred to the box during the process; it is given by:
where
F is the force applied
d is the displacement of the box
Here we have:
F = 87.3 N is the force applied
d = 2.04 m is the displacement of the box
So, the work done to lift the box is:
3)
The power is the rate of work done per unit time. It is calculated as:
where
W is the work done
t is the time taken to do the work
For the child in this problem, we have:
W = 1250 J is the work done by the child running up the stairs
P = 267 W is the power used
Therefore, re-arranging the equation, we find the time taken:
4)
The kinetic energy of an object is the energy possessed by the object due to its motion. Mathematically, it is given by
where
m is the mass of the object
v is its speed
For the rabbit in this problem, we have:
m = 8.642 kg is the mass of the rabbit
KE = 125.6 is its kinetic energy
Solving the formula for v, we find the speed of the rabbit:
5)
The efficiency of a machine is the ratio between the energy produced in output by the machine and the work done in input. Mathematically, it is given by
where
is the energy in output
is the work in input
For the machine in this problem,
is the work in input
is the energy in output
Therefore, the efficiency of this machine is:
6)
During a collision, the total momentum of the system is always conserved before and after the collision. So we can write:
where
is the mass of the first car
is the initial velocity of the first car
is the mass of the 2nd car
is the initial velocity of the 2nd car
is the final velocity of the two cars stuck together (after the collision, they move together)
Solving the equation for v, we find:
7)
The relationship between speed, frequency and wavelength of a wave is given by the wave equation:
where
v is the speed of the wave
f is the frequency of the wave
is the wavelength
For the wave in the string in this problem we have:
(wavelength)
f = 12 Hz (frequency)
So, the speed of the wave is:
8)
The relationship between frequency and wavelength for an electromagnetic wave is given by
where:
c is the speed of light in a vacuum
f is the frequency of the wave
is the wavelength of the wave
For the blue light in this problem, we have
(frequency)
while the speed of light is
So, the wavelength of blue light is:
9)
The sound wave in this problem travels with uniform motion (=constant velocity), therefore we can use the following equation:
where
d is the distance covered by the wave
v is the speed of the wave
t is the time elapsed
In this problem:
v = 343 m/s is the speed of the sound wave
t = 0.287 s is the time elapsed
So, the distance covered by the wave is