An electric toy car with a mass of #4 kg# is powered by a motor with a voltage of #7 V# and a current supply of #9 A#. How long will it take for the toy car to accelerate from rest to #7/2 m/s#?

Question

An electric toy car with a mass of #4 kg# is powered by a motor with a voltage of #7 V# and a current supply of #9 A#.  How long will it take for the toy car to accelerate from rest to #7/2 m/s#?

Charlotte Aaron · Accepted Answer

#0.389 "s"#

The product of voltage and current is the power input.

#P = IV#

#= (7 "V") * (9 "A")#

#= 63 "W"#

The vehicle's change in kinetic energy is

#Delta "KE" = 1/2 m (v^2 - v_0^2)#

#= 1/2 (4 "kg") ((7/2 "m/s")^2 - (0 "m/s")^2)#

#= 24.5 "J"#

The energy divided by the power equals the amount of time required.

#t = frac{Delta "KE"}{P}#

#= frac{24.5 "J"}{63 "W"}#

#= 0.389 "s"#

Carson Alexander · Answer

undefined To find the time it takes for the toy car to accelerate from rest to $ \frac{7}{2} $ m/s, we can use the equation for acceleration:

$$ a = \frac{F}{m} $$

Where:
- $ a $ is the acceleration
- $ F $ is the force
- $ m $ is the mass

We can find the force using the formula for electric power:

$$ P = VI $$

Where:
- $ P $ is the power (in watts)
- $ V $ is the voltage (in volts)
- $ I $ is the current (in amperes)

Given that the toy car is accelerating, the force produced by the motor is equal to the product of the current and the voltage:

$$ F = VI $$

Now, we can rearrange the equation for acceleration to solve for time:

$$ t = \frac{v_f - v_i}{a} $$

Where:
- $ t $ is the time
- $ v_f $ is the final velocity
- $ v_i $ is the initial velocity (0 m/s, as the car starts from rest)
- $ a $ is the acceleration

Given that the final velocity is $ \frac{7}{2} $ m/s, and the initial velocity is 0 m/s, we can substitute these values along with the acceleration found from the force equation into the time equation to find the time it takes for the toy car to accelerate:

$$ t = \frac{\frac{7}{2} - 0}{\frac{VI}{m}} $$

$$ t = \frac{\frac{7}{2}}{\frac{VI}{m}} $$

$$ t = \frac{7m}{2VI} $$

Substitute the given values:

$$ t = \frac{7 \times 4}{2 \times 7 \times 9} $$

$$ t = \frac{28}{126} $$

$$ t \approx 0.222 \, \text{seconds} $$

So, it will take approximately 0.222 seconds for the toy car to accelerate from rest to $ \frac{7}{2} $ m/s.