An electric toy car with a mass of #5 kg# is powered by a motor with a voltage of #12 V# and a current supply of #1 A#. How long will it take for the toy car to accelerate from rest to #3 m/s#?

Question

An electric toy car with a mass of #5 kg# is powered by a motor with a voltage of #12 V# and a current supply of #1 A#.  How long will it take for the toy car to accelerate from rest to #3 m/s#?

Charlotte Aaron · Accepted Answer

#t=1.875s# let the time taken be t s Then electrical energy spent during this time #V*I*t# Rhis energy will be equal to the KE gained=#1/2mv^2# #V*I*t=1/2*m*v^2# #12*1*t=1/2*5*3^2# #t=45/24=15/8=1.875s#

Emily Abbate · Answer

undefined First, calculate the power supplied to the motor using the formula $ P = VI $, where $ V $ is the voltage and $ I $ is the current.

Given:
- $ V = 12 \, \text{V} $
- $ I = 1 \, \text{A} $

$$ P = 12 \, \text{V} \times 1 \, \text{A} = 12 \, \text{W} $$

Next, use the formula for power:
$$ P = F \cdot v $$

Since the car starts from rest, the initial velocity $ u = 0 $.

So, the force ($ F $) can be calculated using Newton's second law:
$$ F = ma $$

Given:
- $ m = 5 \, \text{kg} $
- $ a = \frac{{v - u}}{{t}} = \frac{{3 \, \text{m/s} - 0}}{{t}} = \frac{{3 \, \text{m/s}}}{{t}} $

Substitute $ m $ and $ a $ into the force formula:
$$ F = 5 \, \text{kg} \times \frac{{3 \, \text{m/s}}}{{t}} = \frac{{15 \, \text{N} \cdot \text{m/s}}}{{t}} $$

Since $ P = F \cdot v $:
$$ 12 \, \text{W} = \frac{{15 \, \text{N} \cdot \text{m/s}}}{{t}} \times 3 \, \text{m/s} $$

Solve for $ t $:
$$ t = \frac{{15 \, \text{N} \cdot \text{m/s}}}{{12 \, \text{W}}} = \frac{{15}}{{12}} \, \text{s} $$

$$ t = \frac{{5}}{{4}} \, \text{s} = 1.25 \, \text{s} $$

Therefore, it will take the toy car approximately $ 1.25 \, \text{s} $ to accelerate from rest to $ 3 \, \text{m/s} $.

HIX Tutor · Answer

undefined Great question! Let’s break it down step by step.

1. **Understand the problem:** We need to find out how long it takes for the toy car to go from 0 m/s to 3 m/s.

2. **Find Force:** First, we need to find the force from the motor. Use the formula:
   $$
   \text{Power} = \text{Voltage} \times \text{Current}
   $$
   So,
   $$
   \text{Power} = 12 \, \text{V} \times 1 \, \text{A} = 12 \, \text{W}
   $$

3. **Find Acceleration:** We also know that:
   $$
   \text{Force} = \text{mass} \times \text{acceleration}
   $$
   The power can also be expressed in terms of force and velocity, but we can first find the force directly using Newton's second law.

4. **Force Calculation:** If we rearrange the equations:
   $$
   \text{Force} = \text{mass} \times \text{acceleration}
   $$
   But we want to find acceleration first. Let’s use the power calculated above:
   $$
   \text{Power} = \text{Force} \times \text{Velocity}
   $$
   When the car starts, it has no velocity, but as it accelerates to 3 m/s, we can assume it’s averaging its speed to find average force.

5. **Average Speed:** The average speed from 0 to 3 m/s is:
   $$
   \text{Average Speed} = \frac{0 + 3}{2} = 1.5 \, \text{m/s}
   $$

6. **Find the force that gives this average speed:** 
   $$
   12 \, \text{W} = \text{Force} \times 1.5 \, \text{m/s}
   $$
   So,
   $$
   \text{Force} = \frac{12}{1.5} = 8 \, \text{N}
   $$

7. **Now, find acceleration using Force:** 
   $$
   \text{Force} = 5 \times \text{acceleration}
   $$
   Rearranging gives:
   $$
   \text{acceleration} = \frac{8}{5} = 1.6 \, \text{m/s}^2
   $$

8. **Calculate time to reach 3 m/s:** Use the formula:
   $$
   \text{final speed} = \text{initial speed} + \text{acceleration} \times \text{time}
   $$
   Rearranging gives:
   $$
   \text{time} = \frac{\text{final speed} - \text{initial speed}}{\text{acceleration}} = \frac{3 - 0}{1.6} = 1.875 \, \text{s}
   $$

So, the toy car will take about **1.875 seconds** to reach 3 m/s! Does that make sense?